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Optimization

Many engineering problems are about finding the best: least material, largest volume or lowest cost. Write the quantity as a function of one variable, differentiate and set the derivative to zero. Check that the solution really is a maximum or minimum.

f'(x) = 0

candidates for maxima and minima

f''(x) < 0

maximum

f''(x) > 0

minimum

Symbols

$f(x)$	quantity to optimize
$x$	the variable you can choose

Example

Largest rectangle with perimeter 20: $A = x(10 - x)$ .

$A' = 10 - 2x = 0 \Rightarrow x = 5$ , a square with $A = 25$ .

Also check the end points of the interval. The largest or smallest value may be there.

Practise applications of derivatives for free →

← Complex numbers · Related rates →

Part of Calculus: Applications of derivatives.