Question

Select ALL the intervals where f(x)= -x^3 + 3x^2 +1 is only decreasing.

-infinity < x < 1
5 < x < infinity
-infinity < x < 0
-infinity < x < infinity
2 < x < infinity
0 < x < 2
1 < x < 5

Answer 1

5 < x < ∞

-∞ < x < 0

2 < x < ∞

Explanation:

Given function is
`-x^(3)` + 3
`x^(2)` + 1.

To, Calculate where the function is increasing or decreasing,

we have to calculate the derivative of the function,

so,

f'(x) = -3
`x^(2)` + 6x.

Equating the f'(x)= 0 , we get

x(-3x + 6) = 0

So, f'(x) will be zero at x = 0 and x = 2.

f(x) will be decreasing in the interval where f'(x) will be negative.

now as the coefficient of f'(x) is negative it will be negative between

the interval -∞ < x < 0 and 2 < x < ∞.

Options which lie in these intervals only are

5 < x < ∞

-∞ < x < 0

2 < x < ∞