Subsections

• Background
• Using Maple
• Exercises

Derivatives and Finding Absolute Maxima and Minima on a Closed Interval

Background

The Absolute Extrema of a function on a closed interval is the highest and lowest point on the graph. That is for some value $c$ in the closed interval $[a,b]$, if $f(c)\geq f(x)$ for all $x \in [a,b]$, then $f(c)$ is the absolute maximum value and if $f(c) \leq f(x)$ for all $x \in [a,b]$, then $f(c)$ is the absolute minimum value.

If $f(x)$ is a continuous function on the closed interval $[a,b]$, then there exists numbers $c$ and $d$ in the interval such that $f(c)$ is the global minimum and $f(d)$ is the global maximum. If $f(x)$ is not continuous, then it does not necessarily have an absolute maximum or an absolutr minimum value. One can check for continuity at questionable points by calculating the right and left handed limits. When $f(x)$ is continuous on $[a,b]$, we can find the absolute extrema either at a critical value $c$ where $f'(c)=0$ or $f'(c)$ does not exist or at one of the endpoints $a$ or $b$.

Using Maple

The Maple commands that are most useful are the ones for plotting functions, taking derivatives, and solving equations. By plotting the function and/or its derivatives, you can get a very good idea of approximately where the extrema occur. Then using the solve or fsolve commands you can find the values of $x$ where $f'(x)=0$. Finally, you can use Maple to check the function value at each critical point and endpoint. The example below shows how Maple can be used to find the global extrema for $f(x)=x^3+x^2-8x+1$ on the interval $[-4,2]$.

> f := x-> x^3+x^2-8*x+1;
> plot(f(x),x=-4..2);
> solve(D(f)(x)=0,x);
> evalf(f(4/3));
> f(-2);
> f(-4);
> f(2);

The plot helps to see how many critical values you have. The solve command shows that there are critical values at $x=4/3$ and $x=-2$. Then evaluating the function at each critical value and each endpoint clearly shows that the absolute maximum is $13$ and occurs at the critical value $x=-2$ and the absolute minimum is $-15$ and occurs at the endpoint $-4$.

Exercises

For each function below, find the absolute extrema on the given closed interval if they exist.

1. $$f(x)=x^4-8x^2+4x+2$$ on $$[-\frac{20}{25},\frac{64}{25}]$$.
2. $$f(x)=x^2+4\sin(2x)-1$$ on $[-3,3]$.
3. $$f(x)=\cos(x^3)+\sin(x^3)$$ on $$[\frac{\pi}{2},\frac{\pi}{2}]$$.
4. $$f(x)=-2x^2+\tan(x)$$ on $[0,1.5]$ and $[0,5]$.