MAT137Y1 Lecture Notes - Lecture 13: Maxima And Minima

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10 Feb 2020

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MAT137Y1 Lecture Notes - Lecture 12: Intermediate Value Theorem

MAT137Y1 Lecture Notes - Lecture 13: Maxima And Minima

MAT137Y1 Lecture Notes - Lecture 14: Mean Value Theorem

Document Summary

Given a function f and interval [a, \, b][a,b], the local extrema may be points of discontinuity, points of non- differentiability, or points at which the derivative has value 0. However, none of these points are necessarily local extrema, so the local behavior of the function must be examined for each point. That is, given a point xx, values of the function in the interval (x - c, \, x + c)(x c,x+c) must be tested for sufficiently small cc. Before (a < x) after (a > x) extremum? f(a) < f(x) f(a) > f(x) If a function is continuous, then absolute extrema may be determined according to the following method. Given a function ff and interval [a, \, b][a,b], Determine all critical points of f in the interval [a, \, b][a,b]. Determine the value of ff at each of its critical points.

Related Questions

Find all the critical points of the given function on the given interval. Classify each critical point as a local maximum, a local minimum, a global maximum, a global minimum, or not an extremum. f(x) = sin(x) + cos(x) on [-4pi,4pi]

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If U and V are positive constants, find all critical points of the function. (Enter your answers as a comma-separated list.) F(t) = Ue t + Ve -t t = Find all critical points of f(x) = 20 x 2 e - 0.3x and classify them as local minima of ,maxima. x = (smaller value) This critical point is a . x = (larger value) This critical point is a . Consider f(x) = x 8 (7 - x) 9. Find f '(x) and simplify your answer. f '(x) = Find all critical points of f(x) and then use the first derivation test to determine local maxima and minima. f(x) has a at x = (smallest value) Local Maximum OR Minimum OR Neither f(x) has a at x = Local Maximum OR Minimum OR Neither f(x) has a at x = (largest value) Local Maximum OR Minimum OR Neither Find the inflection points of f(t) = t 4 + t 3 - 18t 2 +8. Give exact answer. t = (smallest value) t = (largest value) Sketch a possible of y = f(x), using the given information about derivation y' = f'(x) and y" = f"(x). Assume that the function is defined and continuous for all real x. Let f be a function f(x) > 0 for all x. Let g = 1/f. If f is increasing in an interval around x 0, what about g? g is increasing in an interval x 0. g is zero in an interval around x 0. g is decreasing in an interval around x 0. We cannot tell whether g is increasing or decreasing in an interval around x 0. g is constant in an interval around x 0. If f has a local maximum at x 1, what about g? g has a local maximum at x 1. g has both a local minimum and a local maximum at x 1. We cannot tell whether g has a local minimum or a local maximum at x 1. g has an inflection point at x 1. g has a local minimum at x 1. If f is a concave down at x 2, what about g? g is increasing at x 2. g is flat at x 2. We cannot tell whether g is concave up of concave down at x 2. g is concave down at x 2. g is concave up at x 2. You are given the graph of the second derivation function f" below. Which of the given have x-values that are inflection points of the function f? (Select all the apply.) A B C D E F G H I none of the above

MAT137Y1 Lecture Notes - Lecture 13: Maxima And Minima

MAT137Y1 Lecture Notes - Lecture 12: Intermediate Value Theorem

MAT137Y1 Lecture Notes - Lecture 13: Maxima And Minima

MAT137Y1 Lecture Notes - Lecture 14: Mean Value Theorem

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