# Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=x+4/x, [0.2,8] absolute minimum value-? absolute maximum value-?

Find the absolute maximum and absolute minimum values of f on the given interval.
$f\left(x\right)=x+\frac{4}{x},\left[0.2,8\right]$
absolute minimum value-?
absolute maximum value-?
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Step 1
We have given
$f\left(x\right)=x+\frac{4}{x},\left[0.2,8\right]$
Step 2
Obtain the critical values
${f}^{\prime }\left(x\right)=1-\frac{4}{{x}^{2}}$
Then
f'(x)=0
$⇒1-\frac{4}{{x}^{2}}=0$
$⇒\frac{{x}^{2}-4}{{x}^{2}}=0$
$⇒\frac{\left(x+2\right)\left(x-2\right)}{{x}^{2}}=0$
$⇒x=-2,2$
Step 3
Since -2 does not lie in the given interval, x = 2 is the only critical number.
Evaluate f(x) at the endpoints and the critical point.
$f\left(0.2\right)=0.2+\frac{4}{0.2}=20.2$
$f\left(2\right)=2+\frac{4}{2}=4$
$f\left(8\right)=8+\frac{4}{8}=8.5$
Step 4
Therefore, the function has
absolute minimum value 4
absolute maximum value 20.2