Caluclus: Table of contents

Maximum and Minimum of a Function

Local maximum and minimum

Consider the function \(f(x)=x^4-2x^2+1\). Without using too much theory we can express \(f\) in the following way: \[f(x)=\left(x^2-1\right)^2=(x-1)^2\cdot (x+1)^2.\] From this expression we see that the function is always non-negative and its minimum \(0\) is attained at \(2\) points: \(-1\) and \(1\). On the real line, the function does not have maximum, because as \(x\) approaches infinity, the value of \(f\) goes to infinity.

Let us now take a look at the graph of the function \(f\):

The graph shows that the minimum is \(0\) at two points: \(-1\) and \(1\). However, the graph shows that the point \(x=0\) is special: the function has a ``peak’’ at that point. It is not real maximum (because we saw that the function goes to \(+\infty\)). This point is called ``local maximum.’’ We are now going to define this term precisely.

Definition (Local maximum) The function \(f\) attains a local maximum at point \(a\) if there is an interval of the form \((p,q)\) that contains \(a\) (i.e. \(p < a < q\)) such that \(f(x)\leq f(a)\) for all \(x\in (p,q)\).

In an analogous way we define the local minimum.

Definition (Local minimum) The function \(f\) attains a local minimum at point \(a\) if there is an interval of the form \((p,q)\) that contains \(a\) (i.e. \(p < a < q\)) such that \(f(x)\geq f(a)\) for all \(x\in (p,q)\).

Application of differential calculus in finding the local extrema

Theorem If the function \(f\) attains local minimum or local maximum at the point \(a\in\mathbb R\), and if \(f\) is differentiable at \(a\), then \(f^{\prime}(a)=0\).

Notice that the previous theorem makes sense: If \(f\) attains maximum or minimum at \(a\), the tangent has to be horizontal at \(a\), hence the slope is \(0\). The previous proof is just a formal way of writing this observation.

Practice problems

Problem 1. Find the critical values for the function \(f(x)=2x^3-3x^2-36x+18\).

Problem 2. Find the point at which the function \(f(x)=x+2\sin x\) attains its minimum on the interval \(\left[\frac{\pi}2,200\right]\).

Problem 3. Find absolute maximum and the absolute minimum of the function \(f(x)=\frac{x^4}4-\frac{x^3}3-2x^2+4x+5\) on the interval \([-3,3]\).

Problem 4. Find the critical values of the function \(f(x)=x^{\frac17}\cdot (2-x)\).

Problem 5. Find the critical values for the function \(f(x)=|x-5|+x^2-2x+7\).