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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions5 Marks
Question
Find the intervals in which the following functions are increasing or decreasing.
$\text{f}(\text{x})=\frac{3}{2}\text{x}^4-4\text{x}^3-45\text{x}^2+51$
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Answer

$\text{f}(\text{x})=\frac{3}{2}\text{x}^4-4\text{x}^3-45\text{x}^2+51$
$f'(x) = 6x^3 - 12x^2 - 90x $
$= 6x(x^2 - 2x - 15) $
$= 6x(x - 5)(x + 3)$
Here, $x = -3, x = 0$
and $x = 5$ are the critical points.
The possible intervals are $(-\infty,-3),(-3,0),(0,5)$ and $(5,\infty)\ ....(1)$
For f(x) to be increasing, we must have
$f'(x) > 0 $
$⇒ 6x(x - 5)(x + 3) > 0$
$[$Since, $6 > 0, 6x(x - 5)(x + 3) > 0 ⇒ x(x - 5)(x + 3) > 0] ⇒ x(x - 5)(x + 3) > 0$
$\Rightarrow\text{x}\in(-3,0)\cup(5,\infty)$
[From eq. 1] So, f(x) is increasing on
$\text{x}\in(-3,0)\cup(5,\infty).$ For f(x) to be decreasing,we must have,
$f'(x) < 0 $
$⇒ 6x(x - 5)(x + 3) < 0$
$[$Since, $6 > 0, 6x(x - 5)(x + 3) < 0 ⇒ x(x - 5)(x + 3) < 0]$
$⇒ x(x - 5)(x + 3) < 0$
$\Rightarrow\text{x}\in(-\infty,-3)\cup(0,5)$
[From eq. 1]So, f(x) is decreasing on $\text{x}\in(-\infty,-3)\cup(0,5).$

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