MATH 121 Lecture 7: Identifying types of critical points, first and second derivative tests, global vs local optimization

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MATH 121 Lecture 6: Families of Functions, Taylor Polynomials, L'hopital's rule

MATH 121 Lecture 7: Identifying types of critical points, first and second derivative tests, global vs local optimization

MATH 121 Lecture Notes - Lecture 8: Integral, Unit, Riemann Sum

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Formalize the first & second derivative test for identifying local maxima & minima. Study generalized optimization problems related to marginal rates. We have found critical pts by looking for pts on the graph where. F"(x) was undefined (although f(x) was defined) We now formalize two ways to determine if a critical pt is a local min, max, or neither. This avoids the need to sketch the graph. One way is the first derivative test: examine the sign of the derivative on opposite sides of the critical pt. Sketch f" sign left of c f" sign right of c. Find the critical points of the function f(x) = 2x3 - 9x2 + 12x + 3. Use the first derivative test to show whether each critical point is a local maximum or a local minimum. Need f"(x) = 6x2 - 18x + 12 (note: set = 0 for crit. pts) 6(x-2)(x-1) 6(x2 - 3x + 2) = 0 (x-2)(x-1) = 0.

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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

48 f(x) = 12 + (x + 1)2 4. (22 pts) The function f, its derivative f', and its second derivative f" are given by the expressions below. f'(x) = -96(x+1) (12+(x + 1)2) (12+(x + 1)2) (a) Circle the correct statement. f"(x)288(x2 + 2x - 3) i. f is even ii. f is odd. iii. f is neither even nor odd. (b) Give an equation for each horizontal asymptote of f. Write "none" if appropriate. Show your work. Show work for parts (c), (d), (e), and (f) in the space below. (c) List the interval(s) on which f is increasing. Write "none" if appropriate. (d) List the interval(s) on which f is decreasing. Write "none" if appropriate (e) List the x-value(s) at which the local minima of f occur. Write "none" if appropriate (f) List the x-value(s) at which the local maxima of f occur. Write "none" if appropriate. (8) List the interval(s) on which f is concave up. Write "none" if appropriate. (h) List the interval(s) on which f is concave down. Write "none" if appropriate. (i) List the x-coordinate(s) of the inflection points of f. Write "none" if appropriate. () Sketch the graph of y = f(x). Mark the locations of any local extrema and/or inflection points with filled-in circles. Your sketch should be consistent with your answers to (a)-0). -10 -9 -8 -7 -6 -5 -4 4 5 6 7 8 9 10

MATH 121 Lecture 7: Identifying types of critical points, first and second derivative tests, global vs local optimization

MATH 121 Lecture 6: Families of Functions, Taylor Polynomials, L'hopital's rule

MATH 121 Lecture 7: Identifying types of critical points, first and second derivative tests, global vs local optimization

MATH 121 Lecture Notes - Lecture 8: Integral, Unit, Riemann Sum

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