Consider the graph of f below. f(x) a Does the function appear to satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? If it does satisfy the nypotheses, does it also satisfy the conclusion? If it does not satisfy the hypotheses, explain why not. Which of the following statements is true? O The function satisfies the hypotheses of the Mean Value Theorem on the interval [a, b] but doesn't satisfy the conclusion of the theorem. O The function doesn't satisfy the hypotheses of the Mean Value Theorem on the interval [a, b] since it is not differentiable on (a, b). The function doesn't satisfy the hypotheses of the Mean Value Theorem on the interval [a, b] since it is not continuous on [a,b]. O The function satisfies both hypotheses and the conclusion of the Mean Value Theorem on the interval [a, b]. O The function doesn't satisfy the hypotheses of the Mean Value Theorem on the interval [a, b] since f' (a) =f' (b).
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.
![Consider the graph of f below.
f (x)
a
Does the function appear to satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? If it does satisfy the
hypotheses, does it also satisfy the conclusion? If it does not satisfy the hypotheses, explain why not. Which of the following
statements is true?
O The function satisfies the hypotheses of the Mean Value Theorem on the interval [a, b] but doesn't satisfy the conclusion
of the theorem.
O The function doesn't satisfy the hypotheses of the Mean Value Theorem on the interval [a, b] since it is not differentiable
on (a, b).
O The function doesn't satisfy the hypotheses of the Mean Value Theorem on the interval [a, b] since it is not continuous on
[a, b].
O The function satisfies both hypotheses and the conclusion of the Mean Value Theorem on the interval [a, b].
O The function doesn't satisfy the hypotheses of the Mean Value Theorem on the interval [a, b] sincef' (a) = f' (b).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a82c709-15d2-4160-acd9-dbda5155610b%2Fe771f222-a7d7-470b-907e-23ae1227e250%2F7r4lwjj_processed.png&w=3840&q=75)
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