#### Question

Find the interval in which the following function are increasing or decreasing f(x) = x^{8} + 6x^{2 }?

#### Solution

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

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

\[f\left( x \right) = x^8 + 6 x^2 \]

\[f'\left( x \right) = 8 x^7 + 12x\]

\[ = 4x \left( 2 x^6 + 3 \right)\]

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

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

\[ \Rightarrow 4x \left( 2 x^6 + 3 \right) > 0 \left[ \text { Since } \left( 2 x^6 + 3 \right) > 0, 4x \left( 2 x^6 + 3 \right) > 0 \Rightarrow x > 0 \right]\]

\[ \Rightarrow x > 0\]

\[ \Rightarrow x \in \left( 0, \infty \right)\]

\[\text { So ,f(x)is increasing on x }\in \left( 0, \infty \right) . \]

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

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

\[ \Rightarrow 4x \left( 2 x^6 + 3 \right) < 0\]

\[ \Rightarrow x < 0 \left[ \text { Since } \left( 2 x^6 + 3 \right) > 0, 4x \left( 2 x^6 + 3 \right) < 0 \Rightarrow x < 0 \right]\]

\[ \Rightarrow x \in \left( - \infty , 0 \right)\]

\[\text { So,f(x)is decreasing on x }\in \left( - \infty , 0 \right) .\]