Question

solve this question with full steps

Answer 1

Differentiate f(X)

f'(X) =
$3{(x + 1)}^{2} {(x - 3)}^{3} + 3 {(x - 3)}^{2} {(x + 1)}^{3}$
Now we take out the common terms

$3 {(x + 1)}^{2} {(x - 3)}^{2}(x + 1 + x - 3) \\ 3 {(x + 1)}^{2} {(x - 3)}^{2}(2x - 2) \\ 6 {(x + 1)}^{2} {(x - 3)}^{2}(x - 1)$
Now you can see that the critical points are

-1, 1 and 3

For X less than -1, we can see that f'(X) attains a negative value, hence slope there is negative.

By doing so further in the next interval of -1 to 1 we see that f'(X) is negative, hence slope there is also negative.

For 1 to 3 and 3 to infinity, slope is positive.

Hence from -∞ to -1, f(X) is decreasing while from -1 to ∞, f(X) is increasing.