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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.
$f(x) = x^3- 12x^2 + 36x + 17$
✓

Answer

$f(x) = x^3- 12x^2 + 36x + 17$
$f'(x) = 3x^2 - 24x + 36$
$= 3(x^2 - 8x + 12)$
$= 3(x - 2)(x - 6)$
For f(x) to be increasing, we must have
$f'(x) > 0$
$\Rightarrow 3(x - 2)(x - 6) > 0$
$\Rightarrow (x - 2)(x - 6) > 0$
$[$Since, $3 > 0, 3(x - 2)(x - 6) > 0 \Rightarrow (x - 2)(x - 6) > 0]$
$\Rightarrow x < 2$ or $x > 6$
$\Rightarrow\text{x}\in(-\infty,2)\cup(6,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,2)\cup(6,\infty).$
For f(x) to be decreasing, we must have,
$f'(x) < 0$
$\Rightarrow 3(x - 2)(x - 6) < 0$
$\Rightarrow (x - 2)(x - 6) < 0$
$[$Since, $3 > 0, 3(x - 2)(x - 6) < 0 \Rightarrow (x - 2)(x - 6) < 0]$
$\Rightarrow 2 < x < 2$
$\Rightarrow\text{x}\in(2,6)$
So, f(x) is decreasing on $\text{x}\in(2,6).$

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