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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Maharashtra 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 $f'(x) > 0$
$\Rightarrow 6x(x - 5)(x + 3) > 0$ [Since, $6 > 0, 6x(x - 5)(x + 3) > 0$
$\Rightarrow x(x - 5)(x + 3) > 0]$
$\Rightarrow x(x - 5)(x + 3) > 0$
$\Rightarrow\text{x}\in(-3,0)\cup(5,\infty)$ [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
$\Rightarrow 6x(x - 5)(x + 3) < 0$ [Since, $6 > 0, 6x(x - 5)(x + 3) < 0$
$\Rightarrow x(x - 5)(x + 3) < 0]$
$\Rightarrow x(x - 5)(x + 3) < 0$
 $\Rightarrow\text{x}\in(-\infty,-3)\cup(0,5)$ [From eq. 1]
So, f(x) is decreasing on $\text{x}\in(-\infty,-3)\cup(0,5).$

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