For each function, find the critical points of f, identify what open intervals is f increasing or decreasing, and identify at what points does f have a local minimum or maximum value. Show all work g(x)= x^2 square root of 5-x
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.
For each function, find the critical points of f, identify what open intervals is f increasing or decreasing, and identify at what points does f have a local minimum or maximum value. Show all work
g(x)= x^2 square root of 5-x
The critical points of a function are the (x,y) points such that, for the x-values of these points, the derivative of the function is either 0 or does not exist. These x-values are called critical values or critical numbers. If a critical value is , then the critical point will be
Once we identify these values we can analyze the first derivative of the function in the intervals between consecutive critical values.
The function will be increasing in the intervals in which the first derivative is positive and
The function will be decreasing In the intervals in which the first derivative is negative.
Further to find the local maximum and minimum values we must evaluate the second derivative of the function at these critical values
The function has a local maximum if and
The function has a local minimum if
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