Mathematics

Problem

(a) State the rule that allows you to decide whether a stationary point of a twice–differentiable real function is a local maximum or a local minimum. (b) A differentiable function

g

satisfies

g'(2)=0

and

g'(5)=0

. Further,

g''(2)=-4

and

g''(5)=9

. (i) Explain what information the given second–derivative values provide about the curvature of the graph at

x=2

and

x=5

. (ii) Hence identify the nature of the stationary points of

g

x=2

and

x=5

. (iii) If

g(2)=7

and

g(5)=3

, write the coordinates of both stationary points. (i) Explain what information the given second–derivative values provide about the curvature of the graph at

x=2

and

x=5

Intuitive explanation

The second derivative test uses the sign of $g''(x)$ at a stationary point to reveal whether the curve is bending upward (local minimum) or downward (local maximum). When $g'(x) = 0$ , a negative second derivative means the tangent line is turning downward—a local maximum—while a positive second derivative means it's turning upward—a local minimum. This test lets you classify stationary points without checking the function's behavior on either side.

Intuitive explanation

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