Find $f ′(x)$ .

The function $f (x)$ has a local maximum at the point where $x = −1$.

Find the value of a.

$f(x) = a{x^2} + 4{x^{ - 1}} - 3$

$f'(x) = 2ax - 4{x^{ - 2}}$     (A3)

(A1) for 2ax, (A1) for –4x ^–2 and (A1) for derivative of –3 being zero.     (C3)

[3 marks]

$2ax - 4x^{-2} = 0$     (M1)

$2a( - 1) - 4{( - 1)^{ - 2}} = 0$     (M1)

$-2a - 4 = 0$

$a = -2$     (A1)(ft)

(M1) for setting derivative function equal to 0. (M1) for inserting $x = -1$ but do not award (M0)(M1)     (C3)

[3 marks]

(a) Many candidates gave up at this point. Those who attempted the derivative did so with varying success. Many could not differentiate a term with a negative index.

(b) In part (b) most substituted the -1 into the original function rather than the differentiated one. They did not realize they had to put the differentiated function equal to zero.