Find the intervals in which the following functions are increasin… - Vidyadip

Question

Find the intervals in which the following functions are increasing or decreasing. $f(x) = 10 - 6x - 2x^2$

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Answer

$f(x) = 10 - 6x - 2x^2$
$f'(x) = -6 - 4x$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow -6 - 4x > 0$
$\Rightarrow -4x > 6$
$\Rightarrow\text{x}<\frac{-3}{2}$
$\Rightarrow\text{x}\in\Big(-\infty,\frac{-3}{2}\Big)$
So,$f(x)$ is increasing on $\Big(-\infty,\frac{-3}{2}\Big).$
For $f(x)$ to be decreasing, we must have
$f'(x) < 0$
$\Rightarrow -6 - 4x < 0$
$\Rightarrow -4x < 6$
$\Rightarrow\text{x}>\frac{-6}{4}$
$\Rightarrow\text{x}>\frac{-3}{2}$
$\Rightarrow\text{x}\in\Big(\frac{-3}{2},\infty\Big)$
So, $f(x)$ is decreasing on $\Big(\frac{-3}{2},\infty\Big).$

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