Question

Let f be the function with derivative f'(x)= x^3 - 3x - 2. Which of the following statements is true?

Answer 1

B. is true

Explanation:

For turning points f'(x) = 0, so

x^3 - 3x - 2 = 0

f(-1) = -1 + 3 - 2 = 0

f(2) = 8 - 6 - 2 = 0

At = -1, f"(x) = 3x^2 - 3 = 3(1) - 3 = 0 so this is a point of inflection

and at x = 2 f"(x) = 3(-2)^2 - 3 = 9 so this is a minimum.

So there ia a point on inflection at x = 0 and a minimum at x = 2

Answer 2

To determine the function f(x), we integrate the given derivative f'(x) as follows:

`∫ f'(x) dx = ∫ (x^3 - 3x - 2) dxf(x) =`

`(1/4)x^4 - (3/2)x^2 - 2x + C`

where C is the constant of integration.

To find the value of C, we can use a point of the function. Let's say that f(0) = 5. Then we have:

`5 = (1/4)(0)^4 - (3/2)(0)^2 - 2(0) + C`

`C = 5`

Therefore, the function f(x) is:

`f(x) = (1/4)x^4 - (3/2)x^2 - 2x + 5`

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