Answer
$f(\frac{\sqrt 3}{3})\approx0.57$ is the absolute maximum value.
$f(0)=0$ is the absolute minimum value.
Work Step by Step
The Closed Interval Method
1) Function $f$ must be continuous on the closed interval $[a, b]$
2) Find the values of $f$ at the critical points in $(a,b)$ and the values of $f$ at the endpoints.
3) Compare the values.
- The largest is the absolute maximum value.
- The smallest is the absolute minimum value.
$$f(t)=\frac{\sqrt t}{1+t^2},\hspace{1.5cm}[0, 2]$$
First, $f(t)$ is continuous on the interval $[0,2]$, so the Closed Interval Method can be applied here.
1) Find critical points of $f$
$$f'(t)=\frac{\frac{1}{2\sqrt t}(1+t^2)-\sqrt t\times2t}{(1+t^2)^2}$$
$$f'(t)=\frac{\frac{1+t^2-2t\sqrt t\times2\sqrt t}{2\sqrt t}}{(1+t^2)^2}$$
$$f'(t)=\frac{1+t^2-4t^2}{2\sqrt t(1+t^2)^2}$$
$$f'(t)=\frac{1-3t^2}{2\sqrt t(1+t^2)^2}$$
We can easily see that $f'(t)$ does not exist when $2\sqrt t(1+t^2)^2=0$, which means $t=0$ or $1+t^2=0$. However, since $1+t^2\gt0$ for all $t$, only $t=0$ is a critical point.
Yet, $t=0$ does not lie in the interval $(0,2)$ given, so it will not be counted here.
Also, $$f'(t)=0$$
$$1-3t^2=0$$
$$t^2=\frac{1}{3}$$
$$t=\pm\frac{\sqrt 3}{3}$$
These values also are critical points. However, only $t=\frac{\sqrt 3}{3}$ lies in the interval $(0,2)$, so only it will be evaluated next.
2) Calculate the values of $f$ at critical points and endpoints.
- Critical points: $t=\frac{\sqrt 3}{3}$
$$f(\frac{\sqrt 3}{3})=\frac{\sqrt{\frac{\sqrt 3}{3}}}{1+(\frac{\sqrt 3}{3})^2}\approx0.57$$
- End points: $t=0$ and $t=2$
$$f(0)=\frac{\sqrt 0}{1+0^2}=0$$
$$f(2)=\frac{\sqrt 2}{1+2^2}=\frac{\sqrt 2}{5}\approx0.283$$
3) Comparison
Among the results:
- $f(\frac{\sqrt 3}{3})\approx0.57$ is the largest, so $f(\frac{\sqrt 3}{3})\approx0.57$ is the absolute maximum value.
- $f(0)=0$ is the smallest, so$f(0)=0$ is the absolute minimum value.
