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Find the absolute maximum and absolute minimum values of $f$ on the given interval: $\;f(x)=\sin x +\cos x ,[0 ,\large\frac {\pi}{3}]$

Note: This is part 6th of a 7 part question, split as 7 separate questions here.

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  • To find the absolute maximum and minimum values of a continuous function f on a closed interval $[a,b]$
  • (i) Find the values of the critical numbers of f in $(a,b)$
  • (ii) Find the value of $f(a)$ and $f(b)$
  • (iii) The largest of the values from (i) and (ii) is the absolute maximun value, the smallest of these values is the absolute minimum value.
$f(x)= \sin x +\cos x$ on $[0, \large\frac{\pi}{3}]$ is continous on $[0, \large\frac{\pi}{3}]$
Step 1:
$f'(x)=\cos x-\sin x$
$f'(x)=0=>\cos x-\sin x=0$
$\tan x =1\; or\; x=\large\frac{\pi}{4} $$ \in [0, \large\frac{\pi}{3}]$
Step 2:
The critical value is $f(\large\frac{\pi}{4})=\frac{1}{\sqrt 2}+\frac{1}{\sqrt 2}=\frac{2}{\sqrt 2}$$=\sqrt 2$
The values of $f$ at the end points are
$f(0)=1 \qquad f(\large\frac{\pi}{3})=\large\frac{\sqrt 3}{2}+\frac{1}{2}=\frac{\sqrt 3+1}{2}$
Step 3:
Comparing the three values , the absolute maximum is $f(\large\frac{\pi}{4})$$=\sqrt 2$ and the absolute minimum is $f(0)=1$
answered Jul 31, 2013 by meena.p

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