Determine the intervals in which the function f (x) = x^4 - 8x^3 … - Vidyadip

Question

Determine the intervals in which the function $f (x) = x^4 - 8x^3 + 22x^2 - 24x + 21$is strictly increasing or strictly decreasing.

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Answer

$f' (x) = 4x^3 - 24x^2 + 44x - 24$
$= 4(x^{3} - 6x^{2} + 11x - 6) = 4(x-1(x - 2) (x - 3)$
$f'(x) = 0 \Rightarrow x = 1, x = 2, x = 3$
The invertible are $(- \infty, 1), (1, 2) (2, 3), (3, \infty)$
$\text{since} f' (x) > 0 \text{ in} (1, 2) \text{and} (3, \infty)$
$\therefore f(x)$ is strictly increasing in $(1, 2) \cup(3, \infty)$
and strictly decreasing in $(-\infty, 1) \cup(2, 3)$

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