Let f (x) = 2x3 + 12x2 + 5.Find the critical numbers for f and use the Second Derivative Test to determine the relative extreme values (if any) of the function. List the critical numbers in numerical order (smallest first). Enter DNE in any unused answer blanks. Critical Number c f '' (c) type of extreme point f (c)     ---Select--- relative maximum relative minimum DNE       ---Select--- relative maximum relative minimum DNE       ---Select--- relative maximum relative minimum DNE

Q: 4z and draw the tangent lines to the graph at points whose x-coordinates Graph the function f(z) =…

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Q: Use the second derivative test to identify any critical point and determine what each critical point…

A: Critical point: The point where the function fx, y is defined and its gradient is a zero vector or…

Q: 5. Suppose the derivative of a function f is f'(x) = (x - 2)²(x + 1)³(x − 8)¹. a) Find the regions…

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Q: Given the derivative f'(x), find the x-value for all relative extrema for the graph of the original…

A: Consider the given derivative. f'x=x2−5x+6 Put the first derivative equal to zero.…

Q: 0. Given the following functions, make a sign sign diagram for • the first and second derivative and…

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Q: The rule of the derivative of a function f is given below (but not the rule of f itself). Find the…

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Q: _f(x) = = x + 1 ; P(-2, 2)

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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 terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

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.

Derivatives

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 one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

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 first derivative test and second derivative test to understand the concave behavior of the function.