Applications of Derivatives

Maxima and Minima

Understand

\(f\) has a local maximum at \(c\) if \(f(c)\ge f(x)\) for all \(x\) near \(c\), and a local minimum if \(f(c)\le f(x)\) near \(c\). At such a point (if \(f\) is differentiable) \(f\,'(c)=0\): \(c\) is a critical point.

Test	Local maximum	Local minimum
First derivative	\(f\,'\) changes from + to −	\(f\,'\) changes from − to +
Second derivative	\(f\,'(c)=0,\ f\,''(c)<0\)	\(f\,'(c)=0,\ f\,''(c)>0\)

If \(f\,''(c)=0\) the second-derivative test fails; use the first-derivative test. A critical point where \(f\,'\) does not change sign is a point of inflection (e.g. \(x^3\) at 0).

Absolute (global) extrema on [a, b]

A continuous function on a closed interval attains an absolute maximum and minimum. Evaluate f at every critical point in (a, b) and at a and b; the largest value is the absolute maximum, the smallest the absolute minimum.

Optimisation problems

Name the quantity to optimise and write it as a function of one variable (use the constraint to eliminate the others).
State the domain.
Differentiate, find critical points, confirm max/min with a test.
Answer the question asked (dimensions, value, units).

Key Concepts

Critical point: f′(c) = 0.
f″(c) < 0 ⇒ local max; f″(c) > 0 ⇒ local min.
Absolute extrema on [a, b]: check critical points and end points.
Optimisation: one-variable function + derivative test.

Formula Bank

Second-derivative test

\[f'(c)=0:\quad f''(c)\lt 0\Rightarrow\text{local max},\qquad f''(c)\gt 0\Rightarrow\text{local min}\]

Key Points

Classic optimisation results

Fixed perimeter ⇒ maximum area rectangle is a square.
Closed cylinder of given volume with least surface area: h = 2r (height = diameter).
Open box from an a × a sheet: cut squares of side a/6.
Sum of two positive numbers fixed ⇒ product is greatest when they are equal.

Absolute extrema on [a, b]

Compare the values at critical points inside the interval and at the two end points.

Common Mistakes

Forgetting the end points

The absolute maximum of \(x^3-3x\) on \([0,3]\) is at the end point \(x=3\) (value 18), not at a critical point.

Local vs absolute

A local maximum need not be the largest value overall; a local minimum value can even exceed a local maximum value.

Solved Examples

Open box of maximum volume

Squares of side \(x\) are cut from the corners of a 30 cm × 30 cm sheet and the sides folded up.

\(V=x(30-2x)^2\), \(V'=(30-2x)(30-6x)=0\Rightarrow x=5\) (\(x=15\) gives \(V=0\)). \(V_{\max}=5\cdot20^2=2000\) cm³.

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Topic Summary

Find critical points, classify them, and for closed intervals compare with the end points. In word problems, reduce to one variable using the constraint.

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