Question

Find the absolute maximum value and the absolute minimum value of the given function in the given intervals $f\left(x\right)=4x-\frac{1}{2}{x}^2,x\in [-2,\frac{9}{2}]$

Answer 1

Given $f\left(x\right)=4x-\frac{1}{2}{x}^2$

Take first derivative of the given function

$f^{{}^{\prime }}\left(x\right)=4-x$

Putting this equal to zero

$f^{{}^{\prime }}\left(x\right)=4-x=0$

$\Rightarrow x=4$

Now let's look at second derivative of the given function.

$f^{{}^{\prime \prime }}\left(x\right)=-1$ and as we can see, this is negative for all $x$; hence $f\left(x\right)$ will have it's maxima at $x=4$.

Maximum value of the given function is

$f\left(4\right)=4\times 4-\frac{1}{2}\times 4^2=8$