Finding maxima and minima

At a maximum or minimum the slope is zero. Whether it is a maximum or a minimum is shown by the curvature, i.e. the second derivative.

Rules

Necessary condition: $f^{'} (x_{0}) = 0$
Sufficient: $f^{''} (x_{0}) < 0$ → maximum, $f^{''} (x_{0}) > 0$ → minimum
If $f^{''} (x_{0}) = 0$ : check the sign change of $f^{'}$
y-value: put $x_{0}$ into $f$

Your turn

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Extrema

Task

Find the extrema of

f (x) = x^{3} - 3 x^{2} - 9 x - 1

$x$ of the maximum

$x$ of the minimum

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Find the extrema of $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ .

Worked solution

1. $f^{'} (x) = 3 x^{2} - 12 x + 9 = 3 (x - 1) (x - 3)$
2.Necessary condition $f^{'} (x) = 0$ : $x_{1} = 1$ , $x_{2} = 3$
3. $f^{''} (x) = 6 x - 12$
4. $f^{''} (1) = - 6$ < 0 → maximum, $f^{''} (3) = 6$ > 0 → minimum

Typical mistake: Mixing up maximum and minimum: $f^{''} (x_{0}) < 0$ means maximum (curve bends down).

Common questions

What if the second derivative is zero?

Then use the sign change of the first derivative: if f' changes from + to −, it is a maximum, from − to +, a minimum, with no change a point of inflection.

Tangent line Power and sum rule Product rule

Finding maxima and minima

Rules

Your turn

What if the second derivative is zero?

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