The second derivative and concavity Problem set B

Find \(f''(2)\) for each function.

1. (a) \(f(x)=9x^2+5x-2\)
2. (b) \(f(x)=3x^2-8x+7\)

Find \(f''(2)\) for each function.

1. (a) \(f(x)=3x^3+2x^2-5x+8\)
2. (b) \(f(x)=x^3-4x^2+9x+6\)

Find \(y''(2)\) for each function. Give your answers to 3 significant figures.

1. (a) \(y=e^{-4x}\)
2. (b) \(y=3e^{2x}\)

Find \(y''(2)\) for each function. Give your answers to 3 significant figures.

1. (a) \(y=4\ln x\)
2. (b) \(y=2-5\ln x\)

Find the values of x for which \(f(x)=x^3-9x^2+7x+4\) is concave-down.

The graph shows part of the curve \(y=-x^3+6x^2-9x+6\). Use the words increasing, decreasing, concave-up and concave-down to describe each section.

1. (a) A to B
2. (b) B to D
3. (c) D to E

The height of a bean plant, h cm, t weeks after planting is modelled by \(h=0.5t^3-6t^2+30t\).

1. (a) Find \(h''(3)\).
2. (b) Say what the sign of your answer tells you about the rate at which the plant is growing at t = 3.

The concentration of a drug in the blood is \(C=60e^{-0.3t}\) mg per litre, t hours after a dose.

1. (a) Find \(C''(2)\) to 3 significant figures.
2. (b) Is the curve concave-up or concave-down? Say what this means for the model.

Find the values of x for which \(f(x)=x^4-12x^2+5\) is concave-down.

Find \(d^2y/dx^2\) for \(y=2e^{3x}-5x\), and explain why the curve is concave-up for every value of x.