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Increasing and Decreasing Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions5 Marks
Question
Find the intervals in which the following functions are increasing or decreasing. $\text{f}(\text{x})=\frac{\text{x}^4}{4}+\frac{2}{3}\text{x}^3-\frac{5}{4}\text{x}^2-6\text{x}+7$
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Answer

$\text{f}(\text{x})=\frac{\text{x}^4}{4}+\frac{2}{3}\text{x}^3-\frac{5}{4}\text{x}^2-6\text{x}+7$
$\therefore f'(x) = x^3 + 2x^2- 5x - 6$
Critical points
$f'(x) = 0$
$\Rightarrow x^3 + 2x^2- 5x - 6 = 0$
$\Rightarrow (x + 1)(x + 3)(x - 2) = 0$
$\Rightarrow x = -1, -3, 2$
Clearly, $f'(x) > 0$ if $-3 x < -1$ and $x > 2$
$f'(x) < 0$ if $x < -3$ and $-1 < x < 2$
Thus, $f(x)$ increases in $(-3,-1)\cup(2,\infty),$ decreases in $(-\infty,-3)\cup(-1,2).$

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