Classifying stationary points and inflection Problem set B

Find the values of x for which:

1. (a) \(f(x)=x^3-12x^2+4x+3\) is concave-down;
2. (b) \(f(x)=x^3+15x^2-2x+8\) is concave-up.

Find the coordinates of the stationary points of \(y=x^3+6x^2-15x+1\), and classify each as a local maximum or a local minimum.

Find the coordinates of the stationary points of \(y=x^4-32x^2-7\), and classify each as a local maximum or a local minimum.

Find the coordinates of the stationary points of \(y=10x^{1/2}-4x+1\), and classify each as a local maximum or a local minimum.

Find the coordinates of the stationary points of \(y=5x-e^x\), and classify each as a local maximum or a local minimum. Give coordinates to 3 significant figures.

Find the coordinates of the stationary points of \(y=4x-6\ln x\), and classify each as a local maximum or a local minimum. Give coordinates to 3 significant figures.

\(f(x)=x^3+kx^2+2x-1\) and \(f''(-2)=-4\). Find the value of k.

\(f(x)=x^3+ax^2+bx-2\), with \(f''(-3)=-14\) and \(f'(2)=13\). Find the values of a and b.

\(y=xe^{-2x}\).

1. (a) Find \(d^2y/dx^2\).
2. (b) Find the value of \(d^2y/dx^2\) when \(x=0\).

A farmer puts x kg of fertiliser on each hectare. The crop yield is \(Y=-x^3+12x^2-36x+50\) tonnes, for \(0< x\le 8\). Find the value of x that gives the maximum, using the second derivative to show that it is a maximum, and state the maximum value.

The number of students who have heard a rumour, R hundred, t days after it starts is modelled by \(R=-t^3+18t^2+5t\) for \(0\le t\le 18\).

1. (a) Find \(R''\).
2. (b) Find the value of t at which \(R''=0\), and show that the concavity changes there.
3. (c) Explain why the rumour is spreading fastest at this time, and find that rate of growth.

Sketch a curve for \(x>0\) where \(f(x)>0\), \(f'(x)<0\) and \(f''(x)>0\).

Find the stationary points of \(y=3x^4-8x^3\). Classify each one. If the second derivative test does not decide, say what you do instead.