5. Use the derivative of the function y = f(x) to find the points at which f has

Question

5.

Use the derivative of the function y = f(x) to find the points at which f has a (a) local maximum, (b) local minimum, or (c) point of inflection. Y' = (x - 3)^2(x - 5) (a) and (b) At what x-values do the local maxima occur, if any, and at what x-values do the local minima occur, if any? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. The the local maxima are at x = ___ and the local minima are at x = ____. (Type an integer or simplified fraction. Use a comma to separate answers as needed.) The local maxima are at x = ___ and there are no local minima. (Type an integer or simplified fraction. Use a comma to separate answers as needed.) There are no local maxima and there are no local minima. At what x-value(s), if any, do the infection points occur? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. x = ____ (Type an integer or simplified fraction. Use a comma to separate answers as needed.) There is no solution.

Explanation / Answer

Given,

y' = (x-3)2(x-5)

finding critical points ->   y' = 0

therefore, (x-3)2(x-5) = 0

or x = 3, 5

y' = x3 - 11x2 + 39x - 45

y" = 3x2 - 22x + 39

y"(3) = 27 - 66 + 39 = 0

Therefore, we have an inflection point x = 3

we can verify this by finding y" for neighbouring points of x = 3 i.e. x = 2 and x = 4

y"(2) = 12 + 39 - 44 = 7 (upward concavity)

and y"(4) = 48 + 39 - 88 = -1 (downward concavity)

Hence, x = 3 is an Inflection point

Now, y"(5) = 75 + 39 - 110 = 4 (upward concavity)

Therefore, at x = 5 we have a minimum

There is no maximum

a) C. x = 5

b) A. x = 3

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