The curve C has equation f(x) = 4(x^1.5) + 48/(x^0.5) - 8^0.5 for x > 0. (a) Find the exact coordinates of the stationary point of C. (b) Determine whether the stationary point is a maximum or minimum.

Part a:f(x) = 4x^1.5 + 48x^-0.5 - 8^0.5Finding the gradient function, f'(x) = 6x^0.5 - 24x^-1.5At the stationary point, the gradient is zero, so f'(x) = 06x^0.5 - 24x^-1.5 = 06x² - 24=0x² = 4x = 2 is the solution. x = -2 is ignored as C is only defined for x > 0.f(2) = 4(2)^1.5 + 48(2)^-0.5 - 8^0.5 = 8(2^0.5) + 24(2^0.5) - 2(2^0.5) = (2^0.5)*(8+24-2) = 30(2^0.5) The stationary point is (2, 30(2^0.5)).===========================================Part b:To analyse concavity, we need the rate of change in the gradient, i.e. the second derivate:f''(x) = 3x^-0.5 + 36x^-2.5f''(2) = 3(2)^-0.5 + 36(2)^-2.5 > 0This means that at the stationary point, the gradient is increasing. The stationary point is a minimum.