A curve C has equation y = x^2 − 2x − 24sqrt x, x > 0. Prove that it has a stationary point at x=4.

A stationary point is where the curve has 0 gradient. So to prove that x=4 is a stationary point, we must find the equation of the first derivative. To do this, differentiate x² - 2x - 24sqrtx. It might help to rewrite all terms as indices: x² -2x -24x^1/2.Now we can differentiate. Differentiate by multiplying by the power and then taking one from the power, to give: 2x - 2 - (24 x 1/2)x^-1/2 which simplifies to 2x - 2 -12x^-1/2Now substitute x=4 in to find the gradient of the curve when x=4: dy/dx = (2 x 4) -2 - 12(4^-1/2) = 8 - 2 - 12/2 = 0 Hence there is a stationary point at x=4 as the gradient here is equal to 0