(i) The function f f\,f is defined by
f(x)=(2x−5)2x+1,x≠−1 f(x) = \frac{(2x - 5)^2}{x + 1}, \quad x \neq -1 f(x)=x+1(2x−5)2,x=−1Find f′(x)f'(x)f′(x) in the form P(x)Q(x)\displaystyle \frac{P(x)}{Q(x)}Q(x)P(x) where P(x)P(x)P(x) and Q(x)Q(x)Q(x) are fully factorised quadratic expressions.
Hence find the set of values of x x\,x for which f(x)f(x)f(x) is increasing.
The curve with equation y=g(x)y = g(x)y=g(x) where
g(x)=sin5xx,0<x<π5 g(x) = \frac{\sin 5x}{\sqrt{x}}, \quad 0 < x < \frac{\pi}{5} g(x)=xsin5x,0<x<5πhas a stationary point at MMM.
Show that the xxx-coordinate of M M\,M satisfies the equation tan5x+kx=0\tan 5x + kx = 0tan5x+kx=0, where k k\,k is a constant to be found.
