Question

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6. If f'(x) = e* (x - 1)(x + 2), what are the critical number(s) of f(x)?
A) -2,0, 1
B) 0
C) -2 and 1
D) -2
E) None of these provide all the critical numbers of f(x)
7. Find the indefinite integral: f 12x¹ dx
x +7
(A) 3 In In x + 7 + C
(B) 3 In In 4x³ + C
(C) In In |4x³+ C
(D) In In x + 7 + C
(E) The integral does not exist.

Answer 1

1. To find the critical numbers of f(x), we need to find the values of x where f'(x) = 0 or where f'(x) is undefined. In this case, f'(x) is defined for all real numbers. Therefore, we need to find where f'(x) = 0:

f'(x) = e*(x - 1)(x + 2) = 0

This equation is satisfied when x = 1 and x = -2, so the critical numbers of f(x) are -2 and 1. Therefore, the answer is (C).

2. To find the indefinite integral of f(x) = 12x/(x+7), we can use u-substitution with u = x + 7:

∫ 12x/(x+7) dx = ∫ (12(u-7))/u du

= ∫ (12u - 84)/u du

= 12∫ 1/u du - 84∫ du

= 12ln|u| - 84ln|u| + C

= (12ln|x+7| - 84ln|x+7|) + C

= (-72ln|x+7|) + C

Therefore, the answer is (E) The integral does not exist.

Explanation: