CALCULUS (CLOTH)
CALCULUS (CLOTH)
4th Edition
ISBN: 9781319050733
Author: Rogawski
Publisher: MAC HIGHER
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Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
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A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
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In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 15.7, Problem 12E
To determine

The critical points of the function

We need to use the Second Derivative Test to determine whether they are local minimum, local maximum or saddle points or state that the test fails.

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Chapter 15.7, Problem 11E
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Chapter 15.7, Problem 13E
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Chapter 15 Solutions

CALCULUS (CLOTH)

Ch. 15.1 - Prob. 1PQCh. 15.1 - Prob. 2PQCh. 15.1 - Prob. 3PQCh. 15.1 - Prob. 4PQCh. 15.1 - Prob. 5PQCh. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5E
Ch. 15.1 - Prob. 6ECh. 15.1 - Prob. 7ECh. 15.1 - Prob. 8ECh. 15.1 - Prob. 9ECh. 15.1 - Prob. 10ECh. 15.1 - Prob. 11ECh. 15.1 - Prob. 12ECh. 15.1 - Prob. 13ECh. 15.1 - Prob. 14ECh. 15.1 - Prob. 15ECh. 15.1 - Prob. 16ECh. 15.1 - Prob. 17ECh. 15.1 - Prob. 18ECh. 15.1 - Prob. 19ECh. 15.1 - Prob. 20ECh. 15.1 - Prob. 21ECh. 15.1 - Prob. 22ECh. 15.1 - Prob. 23ECh. 15.1 - Prob. 24ECh. 15.1 - Prob. 25ECh. 15.1 - Prob. 26ECh. 15.1 - Prob. 27ECh. 15.1 - Prob. 28ECh. 15.1 - Prob. 29ECh. 15.1 - Prob. 30ECh. 15.1 - Prob. 31ECh. 15.1 - Prob. 32ECh. 15.1 - Prob. 33ECh. 15.1 - Prob. 34ECh. 15.1 - Prob. 35ECh. 15.1 - Prob. 36ECh. 15.1 - Prob. 37ECh. 15.1 - Prob. 38ECh. 15.1 - Prob. 39ECh. 15.1 - Prob. 40ECh. 15.1 - Prob. 41ECh. 15.1 - Prob. 42ECh. 15.1 - Prob. 43ECh. 15.1 - Prob. 44ECh. 15.1 - Prob. 45ECh. 15.1 - Prob. 46ECh. 15.1 - Prob. 47ECh. 15.1 - Prob. 48ECh. 15.1 - Prob. 49ECh. 15.1 - Prob. 50ECh. 15.1 - Prob. 51ECh. 15.1 - Prob. 52ECh. 15.1 - Prob. 53ECh. 15.1 - Prob. 54ECh. 15.1 - Prob. 55ECh. 15.1 - Prob. 56ECh. 15.1 - Prob. 57ECh. 15.1 - Prob. 58ECh. 15.1 - Prob. 59ECh. 15.1 - Prob. 60ECh. 15.1 - Prob. 61ECh. 15.2 - Prob. 1PQCh. 15.2 - Prob. 2PQCh. 15.2 - Prob. 3PQCh. 15.2 - Prob. 4PQCh. 15.2 - Prob. 1ECh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5ECh. 15.2 - Prob. 6ECh. 15.2 - Prob. 7ECh. 15.2 - Prob. 8ECh. 15.2 - Prob. 9ECh. 15.2 - Prob. 10ECh. 15.2 - Prob. 11ECh. 15.2 - Prob. 12ECh. 15.2 - Prob. 13ECh. 15.2 - Prob. 14ECh. 15.2 - Prob. 15ECh. 15.2 - Prob. 16ECh. 15.2 - Prob. 17ECh. 15.2 - Prob. 18ECh. 15.2 - Prob. 19ECh. 15.2 - Prob. 20ECh. 15.2 - Prob. 21ECh. 15.2 - Prob. 22ECh. 15.2 - Prob. 23ECh. 15.2 - Prob. 24ECh. 15.2 - Prob. 25ECh. 15.2 - Prob. 26ECh. 15.2 - Prob. 27ECh. 15.2 - Prob. 28ECh. 15.2 - Prob. 29ECh. 15.2 - Prob. 30ECh. 15.2 - Prob. 31ECh. 15.2 - Prob. 32ECh. 15.2 - Prob. 33ECh. 15.2 - Prob. 34ECh. 15.2 - Prob. 35ECh. 15.2 - Prob. 36ECh. 15.2 - Prob. 37ECh. 15.2 - Prob. 38ECh. 15.2 - Prob. 39ECh. 15.2 - Prob. 40ECh. 15.2 - Prob. 41ECh. 15.2 - Prob. 42ECh. 15.2 - Prob. 43ECh. 15.2 - Prob. 44ECh. 15.2 - Prob. 45ECh. 15.2 - Prob. 46ECh. 15.2 - Prob. 47ECh. 15.2 - Prob. 48ECh. 15.2 - Prob. 49ECh. 15.2 - Prob. 50ECh. 15.2 - Prob. 51ECh. 15.3 - Prob. 1PQCh. 15.3 - Prob. 2PQCh. 15.3 - Prob. 3PQCh. 15.3 - Prob. 4PQCh. 15.3 - Prob. 5PQCh. 15.3 - Prob. 1ECh. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.3 - Prob. 7ECh. 15.3 - Prob. 8ECh. 15.3 - Prob. 9ECh. 15.3 - Prob. 10ECh. 15.3 - Prob. 11ECh. 15.3 - Prob. 12ECh. 15.3 - Prob. 13ECh. 15.3 - Prob. 14ECh. 15.3 - Prob. 15ECh. 15.3 - Prob. 16ECh. 15.3 - Prob. 17ECh. 15.3 - Prob. 18ECh. 15.3 - Prob. 19ECh. 15.3 - Prob. 20ECh. 15.3 - Prob. 21ECh. 15.3 - Prob. 22ECh. 15.3 - Prob. 23ECh. 15.3 - Prob. 24ECh. 15.3 - Prob. 25ECh. 15.3 - Prob. 26ECh. 15.3 - Prob. 27ECh. 15.3 - Prob. 28ECh. 15.3 - Prob. 29ECh. 15.3 - Prob. 30ECh. 15.3 - Prob. 31ECh. 15.3 - Prob. 32ECh. 15.3 - Prob. 33ECh. 15.3 - Prob. 34ECh. 15.3 - Prob. 35ECh. 15.3 - Prob. 36ECh. 15.3 - Prob. 37ECh. 15.3 - Prob. 38ECh. 15.3 - Prob. 39ECh. 15.3 - Prob. 40ECh. 15.3 - Prob. 41ECh. 15.3 - Prob. 42ECh. 15.3 - Prob. 43ECh. 15.3 - Prob. 44ECh. 15.3 - Prob. 45ECh. 15.3 - Prob. 46ECh. 15.3 - Prob. 47ECh. 15.3 - Prob. 48ECh. 15.3 - Prob. 49ECh. 15.3 - Prob. 50ECh. 15.3 - Prob. 51ECh. 15.3 - Prob. 52ECh. 15.3 - Prob. 53ECh. 15.3 - Prob. 54ECh. 15.3 - Prob. 55ECh. 15.3 - Prob. 56ECh. 15.3 - Prob. 57ECh. 15.3 - Prob. 58ECh. 15.3 - Prob. 59ECh. 15.3 - Prob. 60ECh. 15.3 - Prob. 61ECh. 15.3 - Prob. 62ECh. 15.3 - Prob. 63ECh. 15.3 - Prob. 64ECh. 15.3 - Prob. 65ECh. 15.3 - Prob. 66ECh. 15.3 - Prob. 67ECh. 15.3 - Prob. 68ECh. 15.3 - Prob. 69ECh. 15.3 - Prob. 70ECh. 15.3 - Prob. 71ECh. 15.3 - Prob. 72ECh. 15.3 - Prob. 73ECh. 15.3 - Prob. 74ECh. 15.3 - Prob. 75ECh. 15.3 - Prob. 76ECh. 15.3 - Prob. 77ECh. 15.3 - Prob. 78ECh. 15.3 - Prob. 79ECh. 15.3 - Prob. 80ECh. 15.4 - Prob. 1PQCh. 15.4 - Prob. 2PQCh. 15.4 - Prob. 3PQCh. 15.4 - Prob. 4PQCh. 15.4 - Prob. 5PQCh. 15.4 - Prob. 6PQCh. 15.4 - Prob. 1ECh. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.4 - Prob. 7ECh. 15.4 - Prob. 8ECh. 15.4 - Prob. 9ECh. 15.4 - Prob. 10ECh. 15.4 - Prob. 11ECh. 15.4 - Prob. 12ECh. 15.4 - Prob. 13ECh. 15.4 - Prob. 14ECh. 15.4 - Prob. 15ECh. 15.4 - Prob. 16ECh. 15.4 - Prob. 17ECh. 15.4 - Prob. 18ECh. 15.4 - Prob. 19ECh. 15.4 - Prob. 20ECh. 15.4 - Prob. 21ECh. 15.4 - Prob. 22ECh. 15.4 - Prob. 23ECh. 15.4 - Prob. 24ECh. 15.4 - Prob. 25ECh. 15.4 - Prob. 26ECh. 15.4 - Prob. 27ECh. 15.4 - Prob. 28ECh. 15.4 - Prob. 29ECh. 15.4 - Prob. 30ECh. 15.4 - Prob. 31ECh. 15.4 - Prob. 32ECh. 15.4 - Prob. 33ECh. 15.4 - Prob. 34ECh. 15.4 - Prob. 35ECh. 15.4 - Prob. 36ECh. 15.4 - Prob. 37ECh. 15.4 - Prob. 38ECh. 15.4 - Prob. 39ECh. 15.4 - Prob. 40ECh. 15.4 - Prob. 41ECh. 15.4 - Prob. 42ECh. 15.4 - Prob. 43ECh. 15.4 - Prob. 44ECh. 15.4 - Prob. 45ECh. 15.4 - Prob. 46ECh. 15.4 - Prob. 47ECh. 15.5 - Prob. 1PQCh. 15.5 - Prob. 2PQCh. 15.5 - Prob. 3PQCh. 15.5 - Prob. 4PQCh. 15.5 - Prob. 5PQCh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15.5 - Prob. 5ECh. 15.5 - Prob. 6ECh. 15.5 - Prob. 7ECh. 15.5 - Prob. 8ECh. 15.5 - Prob. 9ECh. 15.5 - Prob. 10ECh. 15.5 - Prob. 11ECh. 15.5 - Prob. 12ECh. 15.5 - Prob. 13ECh. 15.5 - Prob. 14ECh. 15.5 - Prob. 15ECh. 15.5 - Prob. 16ECh. 15.5 - Prob. 17ECh. 15.5 - Prob. 18ECh. 15.5 - Prob. 19ECh. 15.5 - Prob. 20ECh. 15.5 - Prob. 21ECh. 15.5 - Prob. 22ECh. 15.5 - Prob. 23ECh. 15.5 - Prob. 24ECh. 15.5 - Prob. 25ECh. 15.5 - Prob. 26ECh. 15.5 - Prob. 27ECh. 15.5 - Prob. 28ECh. 15.5 - Prob. 29ECh. 15.5 - Prob. 30ECh. 15.5 - Prob. 31ECh. 15.5 - Prob. 32ECh. 15.5 - Prob. 33ECh. 15.5 - Prob. 34ECh. 15.5 - Prob. 35ECh. 15.5 - Prob. 36ECh. 15.5 - Prob. 37ECh. 15.5 - Prob. 38ECh. 15.5 - Prob. 39ECh. 15.5 - Prob. 40ECh. 15.5 - Prob. 41ECh. 15.5 - Prob. 42ECh. 15.5 - Prob. 43ECh. 15.5 - Prob. 44ECh. 15.5 - Prob. 45ECh. 15.5 - Prob. 46ECh. 15.5 - Prob. 47ECh. 15.5 - Prob. 48ECh. 15.5 - Prob. 49ECh. 15.5 - Prob. 50ECh. 15.5 - Prob. 52ECh. 15.5 - Prob. 53ECh. 15.5 - Prob. 54ECh. 15.5 - Prob. 55ECh. 15.5 - Prob. 56ECh. 15.5 - Prob. 57ECh. 15.5 - Prob. 58ECh. 15.5 - Prob. 59ECh. 15.5 - Prob. 60ECh. 15.5 - Prob. 61ECh. 15.5 - Prob. 62ECh. 15.5 - Prob. 63ECh. 15.5 - Prob. 64ECh. 15.5 - Prob. 65ECh. 15.5 - Prob. 66ECh. 15.5 - Prob. 67ECh. 15.5 - Prob. 68ECh. 15.5 - Prob. 69ECh. 15.5 - Prob. 70ECh. 15.5 - Prob. 71ECh. 15.5 - Prob. 72ECh. 15.5 - Prob. 73ECh. 15.5 - Prob. 74ECh. 15.5 - Prob. 75ECh. 15.5 - Prob. 76ECh. 15.5 - Prob. 77ECh. 15.6 - Prob. 1PQCh. 15.6 - Prob. 2PQCh. 15.6 - Prob. 3PQCh. 15.6 - Prob. 4PQCh. 15.6 - Prob. 5PQCh. 15.6 - Prob. 6PQCh. 15.6 - Prob. 1ECh. 15.6 - Prob. 2ECh. 15.6 - Prob. 3ECh. 15.6 - Prob. 4ECh. 15.6 - Prob. 5ECh. 15.6 - Prob. 6ECh. 15.6 - Prob. 7ECh. 15.6 - Prob. 8ECh. 15.6 - Prob. 9ECh. 15.6 - Prob. 10ECh. 15.6 - Prob. 11ECh. 15.6 - Prob. 12ECh. 15.6 - Prob. 13ECh. 15.6 - Prob. 14ECh. 15.6 - Prob. 15ECh. 15.6 - Prob. 16ECh. 15.6 - Prob. 17ECh. 15.6 - Prob. 18ECh. 15.6 - Prob. 19ECh. 15.6 - Prob. 20ECh. 15.6 - Prob. 21ECh. 15.6 - Prob. 22ECh. 15.6 - Prob. 23ECh. 15.6 - Prob. 24ECh. 15.6 - Prob. 25ECh. 15.6 - Prob. 26ECh. 15.6 - Prob. 27ECh. 15.6 - Prob. 28ECh. 15.6 - Prob. 29ECh. 15.6 - Prob. 30ECh. 15.6 - Prob. 31ECh. 15.6 - Prob. 32ECh. 15.6 - Prob. 33ECh. 15.6 - Prob. 34ECh. 15.6 - Prob. 35ECh. 15.6 - Prob. 36ECh. 15.6 - Prob. 37ECh. 15.6 - Prob. 38ECh. 15.6 - Prob. 39ECh. 15.6 - Prob. 40ECh. 15.6 - Prob. 41ECh. 15.6 - Prob. 42ECh. 15.6 - Prob. 43ECh. 15.6 - Prob. 44ECh. 15.6 - Prob. 45ECh. 15.6 - Prob. 46ECh. 15.6 - Prob. 47ECh. 15.6 - Prob. 48ECh. 15.6 - Prob. 49ECh. 15.6 - Prob. 50ECh. 15.6 - Prob. 51ECh. 15.6 - Prob. 52ECh. 15.6 - Prob. 53ECh. 15.6 - Prob. 54ECh. 15.6 - Prob. 55ECh. 15.7 - Prob. 1PQCh. 15.7 - Prob. 2PQCh. 15.7 - Prob. 3PQCh. 15.7 - Prob. 1ECh. 15.7 - Prob. 2ECh. 15.7 - Prob. 3ECh. 15.7 - Prob. 4ECh. 15.7 - Prob. 5ECh. 15.7 - Prob. 6ECh. 15.7 - Prob. 7ECh. 15.7 - Prob. 8ECh. 15.7 - Prob. 9ECh. 15.7 - Prob. 10ECh. 15.7 - Prob. 11ECh. 15.7 - Prob. 12ECh. 15.7 - Prob. 13ECh. 15.7 - Prob. 14ECh. 15.7 - Prob. 15ECh. 15.7 - Prob. 16ECh. 15.7 - Prob. 17ECh. 15.7 - Prob. 18ECh. 15.7 - Prob. 19ECh. 15.7 - Prob. 20ECh. 15.7 - Prob. 21ECh. 15.7 - Prob. 22ECh. 15.7 - Prob. 23ECh. 15.7 - Prob. 24ECh. 15.7 - Prob. 25ECh. 15.7 - Prob. 26ECh. 15.7 - Prob. 27ECh. 15.7 - Prob. 28ECh. 15.7 - Prob. 29ECh. 15.7 - Prob. 30ECh. 15.7 - Prob. 31ECh. 15.7 - Prob. 32ECh. 15.7 - Prob. 33ECh. 15.7 - Prob. 34ECh. 15.7 - Prob. 35ECh. 15.7 - Prob. 36ECh. 15.7 - Prob. 37ECh. 15.7 - Prob. 38ECh. 15.7 - Prob. 39ECh. 15.7 - Prob. 40ECh. 15.7 - Prob. 41ECh. 15.7 - Prob. 42ECh. 15.7 - Prob. 43ECh. 15.7 - Prob. 44ECh. 15.7 - Prob. 45ECh. 15.7 - Prob. 46ECh. 15.7 - Prob. 47ECh. 15.7 - Prob. 48ECh. 15.7 - Prob. 49ECh. 15.7 - Prob. 50ECh. 15.7 - Prob. 51ECh. 15.7 - Prob. 52ECh. 15.7 - Prob. 53ECh. 15.7 - Prob. 54ECh. 15.7 - Prob. 55ECh. 15.7 - Prob. 56ECh. 15.7 - Prob. 57ECh. 15.7 - Prob. 58ECh. 15.7 - Prob. 59ECh. 15.7 - Prob. 60ECh. 15.7 - Prob. 61ECh. 15.7 - Prob. 62ECh. 15.7 - Prob. 63ECh. 15.7 - Prob. 64ECh. 15.8 - Prob. 1PQCh. 15.8 - Prob. 2PQCh. 15.8 - Prob. 3PQCh. 15.8 - Prob. 1ECh. 15.8 - Prob. 2ECh. 15.8 - Prob. 3ECh. 15.8 - Prob. 4ECh. 15.8 - Prob. 5ECh. 15.8 - Prob. 6ECh. 15.8 - Prob. 7ECh. 15.8 - Prob. 8ECh. 15.8 - Prob. 9ECh. 15.8 - Prob. 10ECh. 15.8 - Prob. 11ECh. 15.8 - Prob. 12ECh. 15.8 - Prob. 13ECh. 15.8 - Prob. 14ECh. 15.8 - Prob. 15ECh. 15.8 - Prob. 16ECh. 15.8 - Prob. 17ECh. 15.8 - Prob. 18ECh. 15.8 - Prob. 19ECh. 15.8 - Prob. 20ECh. 15.8 - Prob. 21ECh. 15.8 - Prob. 22ECh. 15.8 - Prob. 23ECh. 15.8 - Prob. 24ECh. 15.8 - Prob. 25ECh. 15.8 - Prob. 26ECh. 15.8 - Prob. 27ECh. 15.8 - Prob. 28ECh. 15.8 - Prob. 29ECh. 15.8 - Prob. 30ECh. 15.8 - Prob. 31ECh. 15.8 - Prob. 32ECh. 15.8 - Prob. 33ECh. 15.8 - Prob. 34ECh. 15.8 - Prob. 35ECh. 15.8 - Prob. 36ECh. 15.8 - Prob. 37ECh. 15.8 - Prob. 38ECh. 15.8 - Prob. 39ECh. 15.8 - Prob. 40ECh. 15.8 - Prob. 41ECh. 15.8 - Prob. 42ECh. 15.8 - Prob. 43ECh. 15.8 - Prob. 44ECh. 15.8 - Prob. 45ECh. 15.8 - Prob. 46ECh. 15.8 - Prob. 47ECh. 15.8 - Prob. 48ECh. 15.8 - Prob. 49ECh. 15.8 - Prob. 50ECh. 15.8 - Prob. 51ECh. 15.8 - Prob. 52ECh. 15.8 - Prob. 53ECh. 15.8 - Prob. 54ECh. 15.8 - Prob. 55ECh. 15.8 - Prob. 56ECh. 15 - Prob. 1CRECh. 15 - Prob. 2CRECh. 15 - Prob. 3CRECh. 15 - Prob. 4CRECh. 15 - Prob. 5CRECh. 15 - Prob. 6CRECh. 15 - Prob. 7CRECh. 15 - Prob. 8CRECh. 15 - Prob. 9CRECh. 15 - Prob. 10CRECh. 15 - Prob. 11CRECh. 15 - Prob. 12CRECh. 15 - Prob. 13CRECh. 15 - Prob. 14CRECh. 15 - Prob. 15CRECh. 15 - Prob. 16CRECh. 15 - Prob. 17CRECh. 15 - Prob. 18CRECh. 15 - Prob. 19CRECh. 15 - Prob. 20CRECh. 15 - Prob. 21CRECh. 15 - Prob. 22CRECh. 15 - Prob. 23CRECh. 15 - Prob. 24CRECh. 15 - Prob. 25CRECh. 15 - Prob. 26CRECh. 15 - Prob. 27CRECh. 15 - Prob. 28CRECh. 15 - Prob. 29CRECh. 15 - Prob. 30CRECh. 15 - Prob. 31CRECh. 15 - Prob. 32CRECh. 15 - Prob. 33CRECh. 15 - Prob. 34CRECh. 15 - Prob. 35CRECh. 15 - Prob. 36CRECh. 15 - Prob. 37CRECh. 15 - Prob. 38CRECh. 15 - Prob. 39CRECh. 15 - Prob. 40CRECh. 15 - Prob. 41CRECh. 15 - Prob. 42CRECh. 15 - Prob. 43CRECh. 15 - Prob. 44CRECh. 15 - Prob. 45CRECh. 15 - Prob. 46CRECh. 15 - Prob. 47CRECh. 15 - Prob. 48CRECh. 15 - Prob. 49CRECh. 15 - Prob. 50CRECh. 15 - Prob. 51CRECh. 15 - Prob. 52CRECh. 15 - Prob. 53CRECh. 15 - Prob. 54CRECh. 15 - Prob. 55CRECh. 15 - Prob. 56CRECh. 15 - Prob. 57CRECh. 15 - Prob. 58CRECh. 15 - Prob. 59CRECh. 15 - Prob. 60CRECh. 15 - Prob. 61CRECh. 15 - Prob. 62CRECh. 15 - Prob. 63CRE
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Finding Local Maxima and Minima by Differentiation; Author: Professor Dave Explains;https://www.youtube.com/watch?v=pvLj1s7SOtk;License: Standard YouTube License, CC-BY