MATH 1080 Lecture Notes - Lecture 17: Inflection Point, Cubic Function, Inflection

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19 Nov 2017

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Page 225 (what you will need for the final exam: for the function of f(x) = 2x3 2x3 12x. Find critical points but need to find the critical numbers first. F" (x) = 6x2 - 6x 12 = 0. = 6(x2 x 2) = 0. = 6(x-2) (x+1) = 2,1 f(2) = 16-12-24 2, 20** f(-1) = -2 3 12 -1, 17** F"" (x) = 0 2x = 6 x = (this is a possible inflection point) Forever concave down all the way to the left. There are no wobbles in the graph due to concavity. When we did the first derivative we didn"t know whether it did a large concavity or not. This is what we are going to study next. Ex2: f (x) 3x4 -4x3 (find all critical points and local extrema) F(x) = x3 (3x 4) x = 0 and x = 4/3 (those are your intercepts) (0,0) (4/3, 0)

Related Questions

Suppose that the first and second derivatives of a function y=f(x) are given by: f'(x) = xe^x and f"(x) = e^x (x+1). Which of the following statements is true? a) f'(-1) > 0 b) f(x) is increasing and opens up (0, infinity) c) There are no critical points and there is one inflection point. d) f(x) is increasing and opens down on (negative infinity, -1) e) There are is one critical point and there are two inflection points. f) f"(x) > 0 on (negative infinity, -1) g) There are no critical points and there are two inflection points. h) f'(x) > 0 on (negative infinity, 0) i) f'(x) < 0 on (0, infinity) j) f"(x) < 0 on (-1, infinity)

f(x) = xe^-x^2

PLEASE SHOW WORK

a) find all critical numbers

b) find intervals that f is increasing/decreasing

c) use first derivative test for local max and min

d) apply second derivative test to f or state that it fails

e) find intervals on f for concave up and concave down

f) find all inflection points

g) Use L'Hopital's rule to find lim as x goes to infinity of f(x) and lim as x goes to -infinity of f(x)

h) find min and max value of f on interval [0,2]

i) sketch a graph of h

MATH 1080 Lecture Notes - Lecture 17: Inflection Point, Cubic Function, Inflection

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