CAIE IGCSE Additional Math 14.9 derivative tests for maxima and minima

Use this derivative-tests page to evaluate f''(x) or sign changes and justify whether each stationary point is a maximum or minimum.

Exam points

Use f''(x) at each stationary x-value to decide whether it is maximum or minimum.
State the sign of the second derivative and link it to the nature of the point.
Write a complete conclusion explaining why the stationary value is a maximum or minimum.

[Maximum number: 2]

A circle, centre O and radius $r\text{ cm}$ , has a sector OAB of fixed area $10\text{ cm}^2$ . Angle AOB is $\theta$ radians and the perimeter of the sector is $P\text{ cm}$ .

Determine the nature of this stationary value.

$\frac{d^2P}{dr^2}=\frac{40}{r^3}$
As r is positive, $\frac{d^2P}{dr^2}$ is positive, so the stationary value is a minimum.
M1 for complete method; A1 for correct conclusion.

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14.9. Use derivative tests for maxima and minima question 1