#### Question

Find the interval in which the following function are increasing or decreasing f(x) = x^{4} − 4x ?

#### Solution

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

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

\[f\left( x \right) = x^4 - 4x\]

\[f'\left( x \right) = 4 x^3 - 4\]

\[ = 4\left( x^3 - 1 \right)\]

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

\[f'\left( x \right) > 0\]

\[ \Rightarrow 4\left( x^3 - 1 \right) > 0 \]

\[ \Rightarrow x^3 - 1 > 0\]

\[ \Rightarrow x^3 > 1\]

\[ \Rightarrow x > 1\]

\[ \Rightarrow x \in \left( 1, \infty \right)\]

\[\text { So,}f(x)\text { is increasing on }\left( 1, \infty \right) . \]

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

\[f'\left( x \right) < 0\]

\[ \Rightarrow 4\left( x^3 - 1 \right) < 0\]

\[ \Rightarrow x^3 - 1 < 0\]

\[ \Rightarrow x^3 < 1\]

\[ \Rightarrow x < 1\]

\[ \Rightarrow x \in \left( - \infty , 1 \right)\]

\[\text { So,}f(x)\text { is decreasing on }\left( - \infty , 1 \right).\]