Question-43
Consider a function \(\displaystyle f:[ -2,4]\rightarrow \mathbb{R}\) defined as \(\displaystyle f( x) =x^{3} +3x\). If the global maximum value of the function is \(\displaystyle a\) and the global minimum value of the function is \(\displaystyle b\), find \(\displaystyle a+b\).
Does the function have any critical points?
\(62\)
We see that \(\displaystyle f^{\prime }( x) =3x^{2} +3\). This is non-zero in the interval \(\displaystyle [ -2,4]\). The function doesn’t have any critical points in this interval. In fact, the function is strictly increasing in the interval. Therefore, the minimum has to occur at \(\displaystyle x=-2\) and the maximum has to occur at \(\displaystyle x=4\). The required result is. \[ f( -2) +f( 4) =-14+76=\boxed{62} \]