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Finding Critical Point Functions

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find all of the critical points and local maximums and minimums of each function

f(x)=〖2x〗^2-12x+7

find all critical points and local extremes of each function on the given intervals.

f(x)=X^2-3x+5 on the entire real number line.

find all critical points and local extremes of each function on the given intervals.

f(x)=□(1/(x^2+1)) on the entire real number line.

verify that the hypotheses of Rolle's Theorem are satisfied for each of the functions on the given intervals, and find the value of the number(s) "c" that Rolle's Theorem promises.

F(x) =x^2 on [-2, 2]

verify that the hypotheses of the Mean Value Theorem are satisfied for each of the functions on the given intervals, and find the number(s) "c" that the Mean Value Theorem guarantees.

f(x)=sin⁡(x) on [0, π/2]

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1. find all of the critical points and local maximums and minimums of each function

Solution:

To find critical points, put f'(x) = 0

4x-12 = 0
4x = 12
x = 3

Critical point is x = 3

Find f''(x)

Since f''(3) > 0 so function has local minimum at x = 3

Local minimum is at (3, -11).

2. Find all critical points and local extremes of each function on the given intervals.

on the entire real number line.

Solution:

To find critical points, put f'(x) ...

Solution Summary

This posting includes step by step instructions on finding local maximum/minimum of a function. It also explains how to apply Rolle's Theorem and Mean value Theorem.

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