#### Question

Find the interval in which the following function are increasing or decreasing f(x) = 10 − 6x − 2x^{2 }?

#### Solution

\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with } a < b, x < a \text { or } x>b.\].

\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]

\[f(x) = 10 - 6x - 2 x^2 \]

\[f'(x) = - 6 - 4x\]

\[\text { For } f(x) \text { to be increasing, we must have } \]

\[f'(x) > 0\]

\[ \Rightarrow - 6 - 4x > 0\]

\[ \Rightarrow - 4x > 6\]

\[ \Rightarrow x < \frac{- 3}{2}\]

\[ \Rightarrow x \in \left( - \infty , \frac{- 3}{2} \right)\]

\[\text { So }, f(x) \text { is increasing on } \left( - \infty , \frac{- 3}{2} \right) . \]

\[\text { For } f(x) \text { to be decreasing, we must have } \]

\[f'(x) < 0\]

\[ \Rightarrow - 6 - 4x < 0\]

\[ \Rightarrow - 4x < 6\]

\[ \Rightarrow x > \frac{- 6}{4}\]

\[ \Rightarrow x > \frac{- 3}{2}\]

\[ \Rightarrow x \in \left( \frac{- 3}{2}, \infty \right)\]

\[\text { So }, f(x) \text { is decreasing on } \left( \frac{- 3}{2}, \infty \right) .\]