Question
Find the intervals in which the function $\frac{x^4}{4}-x^3-5 x^2+24 x+12$ is (a) strictly increasing (b) strictly decreasing.
✓
Answer
$f(x)=\frac{x^4}{4}-x^3-5 x^2+24 x+12$
$f^{\prime}(x)=x^3-3 x^2-10 x+2$
= (x - 2)(x - 4)(x + 3)
$f^{\prime}(x)=0 \Rightarrow x=-3,2,4$
sign of f'(x) :
$\therefore f(x)$ is strictly increasing on $(-3,2) \cup(4, \infty)$ and $f(x)$ is strictly decreasing on $(-\infty,-3) \cup(2,4)$
$f^{\prime}(x)=x^3-3 x^2-10 x+2$
= (x - 2)(x - 4)(x + 3)
$f^{\prime}(x)=0 \Rightarrow x=-3,2,4$
sign of f'(x) :
$\therefore f(x)$ is strictly increasing on $(-3,2) \cup(4, \infty)$ and $f(x)$ is strictly decreasing on $(-\infty,-3) \cup(2,4)$
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