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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions4 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) = x3 + 2x2 - 5x - 6
Critical points
f'(x) = 0
⇒ x3 + 2x2 - 5x - 6 = 0
⇒ (x + 1)(x + 3)(x - 2) = 0
⇒ 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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