Chapter Review 7: Differentiation

Given that \(y = x^{\frac{3}{2}} + \frac{48}{x}, x > 0\)

a) Find the value of \(x\) and the value of \(y\) when \(\frac{dy}{dx} = 0\). (5 marks)

b) Show that the value of \(y\) which you found in part a) is a minimum. (2 marks)

A curve has equation \(y = x^3 - 5x^2 + 7x - 14\). Determine, by calculation, the coordinates of the stationary points of the curve.

The diagram shows part of the curve with equation \(y = f(x)\), where:

\(f(x) = 200 - \frac{250}{x} - x^2, x > 0\)

The curve cuts the x-axis at the points A and C. The point B is the maximum point of the curve.

a) Find \(f'(x)\). (3 marks)

b) Use your answer to part a) to calculate the coordinates of B. (4 marks)

The diagram shows the part of the curve with equation \(y = 5 - \frac{1}{2}x^2\) for which \(y > 0\).

The point P(x, y) lies on the curve and O is the origin.

a) Show that \(OP^2 = \frac{1}{4}x^4 - 4x^2 + 25\). (3 marks)

b) Find the values of \(x\) for which \(f'(x) = 0\). (4 marks)

c) Hence, find the minimum distance from O to the curve. (4 marks)

A wire is bent into the plane shape ABCDE as shown. Shape ABDE is a rectangle and BCD is a semicircle with diameter BD.

The area of the region enclosed by the wire is \(R \text{ m}^2\), AE = x metres, and AB = ED = y metres. The total length of the wire is 2 m.

a) Find an expression for y in terms of x. (3 marks)

b) Prove that \(R = \frac{x}{8}(8 - 4x - \pi x)\). (4 marks)

c) Find the maximum value of R. (5 marks)