A curve y = f(x) is such that f'(x) = (2x+5)⁻½. The curve has gradient ²⁄₃ at the point (2, 2).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [2] M1 for k(2x+5)^(1/2)
2. Dep M1 for use of f'(x)=2/3 and x=2 in their f'(x) in the form k(2x+5)^(-1/2) + c. Allow term in (2x+5)^(-1/2) unsimplified. Need to see attempt at substitution.
3. [2] M1 for p(2x+5)^(1/2)
4. Dep M1 for use of x=2 and y=2 in their f(x) in the form rx + p(2x+5)^(1/2) + d. Allow term in (2x+5)^(1/2) unsimplified. Need to see an attempt at substitution.
5. [2] M1 for solution of their f'(x)=0, must be in the form p(2x+5)^(-1/2) + q to obtain a value for x.
6. A1 y=0
7. [1] B1 Need to see evidence of substitution for either test. B0 if x=-2 obtained incorrectly in part (a)