# Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: $(iii)\: f (x) =4x-\large\frac{x^2}{2}$$, x \in [– 2, \large\frac{9}{2}] This is third part of multipart q5 ## 1 Answer Toolbox: • \large\frac{d}{dx}(x^n)$$=nx^{n-1}$
Step 1:
On differentiating with respect to x we get
$f'(x)=4-\large\frac{1}{2}$$.2x \qquad=4-\large\frac{2x}{2} \qquad=4-x For extreme values f'(x)=0 4-x=0 x=4 Step 2: Now we find the values of f(x) at x=-2,4,\large\frac{9}{2} f(-2)=4(-2)-\large\frac{(-2)^2}{2} \qquad=-8-\large\frac{4}{2} \qquad=\large\frac{-16-4}{2} \qquad=\large\frac{-20}{2} \qquad=-10 Step 3: f(4)=4(4)-\large\frac{(4)^2}{2} \qquad=16-\large\frac{16}{2} \qquad=\large\frac{32-16}{2}$$=8$
Step 4:
$f(\large\frac{9}{2})=$$4\times \large\frac{9}{2}-\large\frac{1}{2}$$\times \large \frac{81}{4}$
$\qquad=18-\large\frac{81}{8}$
$\qquad=\large\frac{144-81}{8}$
$\qquad=\large\frac{63}{8}$
$\qquad=7.875$
Step 5:
At $x=4$,absolute maximum value$=8$
At $x=2$, absolute minimum value $=-10$
edited Aug 30, 2013