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

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Solution

Increasing: $(- \infty, - 3 / 2)$
Decreasing: $(- 3 / 2, \infty)$
We have, $f (x) = 10 - 6 x - 2 x^{2}$
$\Rightarrow$ $f^{'} (x) = - 6 - 4 x = - 2 (2 x + 3)$
For $f (x)$ to be increasing, we must have
$f^{'} (x) > 0 \Rightarrow - 2 (2 x + 3) > 0 \Rightarrow 2 x + 3 < 0 \Rightarrow x < - \frac{3}{2}$
So, $f (x)$ is increasing on $(- \infty, - 3 / 2)$

For $f (x)$ to be decreasing, we must have
$f^{'} (x) < 0 \Rightarrow - 2 (2 x + 3) < 0 \Rightarrow 2 x + 3 > 0 \Rightarrow x > - \frac{3}{2}$
So, $f (x)$ is decreasing on $(- 3 / 2, \infty)$

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