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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.
$\text{f}(\text{x})=\frac{3}{2}\text{x}^4-4\text{x}^3-45\text{x}^2+51$
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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) = 6x3 - 12x2 - 90x

= 6x(x2 - 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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