Question

Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=x+\frac{4}{x},[0.2,8]

Functions
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asked 2021-05-09
Find the absolute maximum and absolute minimum values of f on the given interval.
\(f(x)=x+\frac{4}{x},[0.2,8]\)

Answers (1)

2021-05-10
Step 1
We have given
\(f(x)=x+\frac{4}{x},[0.2,8]\)
Step 2
Obtain the critical values
\(f'(x)=1-\frac{4}{x^{2}}\)
Then
f'(x)=0
\(\Rightarrow 1-\frac{4}{x^{2}}=0\)
\(\Rightarrow \frac{x^{2}-4}{x^{2}}=0\)
\(\Rightarrow \frac{(x+2)(x-2)}{x^{2}}=0\)
\(\Rightarrow x=-2,2\)
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.
\(f(0.2)=0.2+\frac{4}{0.2}=20.2\)
\(f(2)=2+\frac{4}{2}=4\)
\(f(8)=8+\frac{4}{8}=8.5\)
Step 4
Therefore, the function has
absolute minimum value 4
absolute maximum value 20.2
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