Question
Find the intervals in which the function $f(x) = x^3 - 12x^2 + 36x +17$ is $(a)$ increasing, $(b)$ decreasing.
✓
Answer
$f'(x) = 3x^2 - 24x + 36 = 3(x^2 - 8x + 12) = 3(x - 2) (x - 6)$
$f'(x) = 0 \Rightarrow x = 2, x = 6$
$\therefore$ Possible intervals are $(-\infty, 2), (2, 6), (6, \infty)$
Since $f' (x) > 0$ in $(-\infty, 2), (6, \infty)$
$\therefore f(x)$ is increasing in $(-\infty, 2),\cup (6, \infty)$
And $f' (x) < 0$ in $(2, 6) \therefore f(x)$ is decreasing in $(2, 6).$
$f'(x) = 0 \Rightarrow x = 2, x = 6$
$\therefore$ Possible intervals are $(-\infty, 2), (2, 6), (6, \infty)$
Since $f' (x) > 0$ in $(-\infty, 2), (6, \infty)$
$\therefore f(x)$ is increasing in $(-\infty, 2),\cup (6, \infty)$
And $f' (x) < 0$ in $(2, 6) \therefore f(x)$ is decreasing in $(2, 6).$
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