For a curve of gradient dy/dx = (2/(x^2))-x/4, determine a) d^2y/dx^2 b) the stationary point where y=5/2 c) whether this is a maximum or minmum point and d) the equation of the curve

a) Differentiating gives d²y/dx²=-4x^-3-1/4

b) Let dy/dx=0 and rearrange to find x=2

c) Inserting x=2 into d²y/dx²=-4x^-3-1/4 will show that d²y/dx² is smaller than zero so this is a maximum stationary point.

d) To find the original equation of the curve, dy/dx must be intetgrated which gives y=-2x^-1-x²/8+c

Substituting in x=2 when y=5/2 gives 2.5=-1-0.5+c

Rearrange to give c=4

So the final equation is y=-2x^-1-x²/8+4