The second derivative and concavity Problem set A

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

1. (a) \(f(x)=7x^2-4x+9\)
2. (b) \(f(x)=4x^2+6x-3\)

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

1. (a) \(f(x)=4x^3-x^2+6x+2\)
2. (b) \(f(x)=x^3+5x^2-8x+11\)

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

1. (a) \(y=e^{-3x}\)
2. (b) \(y=5e^{2x}\)

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

1. (a) \(y=6\ln x\)
2. (b) \(y=1-3\ln x\)

Find the values of x for which \(f(x)=x^3-6x^2+4x+1\) is concave-down.

The graph shows part of the curve \(y=x^3-6x^2+9x\). 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

Water runs into a tank. The volume of water, V litres, t minutes after a tap is opened is modelled by \(V=-0.5t^3+6t^2+20t\).

1. (a) Find \(V''(2)\).
2. (b) Say what the sign of your answer tells you about the rate at which the volume is increasing at t = 2.

A car bought new is worth \(V=20\,000e^{-0.2t}\) dollars after t years.

1. (a) Find \(V''(2)\) to 2 decimal places.
2. (b) Is the curve concave-up or concave-down? Say what this means for the model.

Find the values of x for which \(f(x)=x^4-8x^2+3\) is concave-down.

Find \(d^2y/dx^2\) for \(y=4e^{-x}+3x\), and explain why the curve is concave-up for every value of x.