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Increasing and Decreasing Functions — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions5 Marks
Question
Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 6 - 9x - x^2$
✓

Answer

We have,
$f(x) = 6 - 9x - x^2$
$f'(x) = -2x - 9$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow -2x - 9 > 0$
$\Rightarrow -2x > -9$
$\Rightarrow\text{x}<\frac{-9}{2}$
$\Rightarrow\text{x}\in\Big(-\infty,\frac{-9}{2}\Big)$
So, $f(x)$ is increasing on $\Big(-\infty,\frac{-9}{2}\Big).$
For $f(x)$ to be decreasing, we must have
$f'(x) < 0$
$\Rightarrow -2x - 9 < 0$
$\Rightarrow -2x < -9$
$\Rightarrow\text{x}>\frac{-9}{2}$
$\Rightarrow\text{x}\in\Big(\frac{-9}{2},\infty\Big)$
So, $f(x)$ is decreasing on $\Big(\frac{-9}{2},\infty\Big).$

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