Answer to Question #147655 in Calculus for liam donohue

Question

Suppose that
f(x)=x3−6x2+10.
(A) List the x values of all local maxima of f. If there are no local maxima, enter 'NONE'.
x values of local maximums =
(B) List the x values of all local minima of f. If there are no local minima, enter 'NONE'.
x values of local minimums =
(C) List the x values of all the inflection points of f. If there are no inflection points, enter 'NONE'.
x values of inflection points =

Answer 1

$f(x) = x^3-6x^2+10,\\ f'(x) = 3x^2 - 12x.$

We should determine points where $f'(x) = 0.$

$3x^2 - 12x = 0, \\ 3x\cdot(x-4) = 0 \Leftrightarrow x = 0 \; \text{or} \; x = 4.$

Let us determine the sign of derivative:

$f'(x) > 0 \;\; \text{if}\;\; x < 0, \\ f'(x) < 0 \;\; \text{if}\;\; 0< x < 4, \\ f'(x) > 0 \;\; \text{if}\;\; x >4.$

Therefore,

$f(x) \nearrow \;\; \text{if}\;\; x < 0, \\ f(x) \searrow 0 \;\; \text{if}\;\; 0< x < 4, \\ f(x) \nearrow 0 \;\; \text{if}\;\; x >4.$

Therefore, x = 0 is the point of local maximum and x = 4 is the point of local minimum.

(A) x = 0

(B) x = 4

(C) NONE