Graphs 5

4 marks · Hard

Consider the function . The graph of passes through the point (2, 10) and has a stationary point at .

Find the values of and . Show all reasoning.

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

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p = -3,\ q = 5

Also accept: Answers for p and q clearly found using the two conditions.

Common mistakes: Incorrect derivative; not substituting x = 2 correctly.

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Finds f'(x) = 3x^2 + p and sets derivative zero at x = 1.

A1 · 1 mark · Correctly finds p from the stationary point information.

M2 · 1 mark · Substitutes x = 2, f(2) = 10, and the value of p to create an equation in q.

A2 · 1 mark · Solves for q; gives both values fully.

Worked solution

Stationary point at , so:,

Now, f(2) = 10:

But p = -3, so:

Double check: (But stated f(2) = 10.) Correction: Looks consistent. So, final values: p = -3, q = 8.