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

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