Caluclus: Table of contents
# Maximum and Minimum of a Function

## Local maximum and minimum

**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 \(x\in (p,q)\).
**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 \(f^{\prime}(a)=0\).
## 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 \(f(x)=|x-5|+x^2-2x+7\).

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.

In an analogous way we define the local minimum.

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.