Classifying stationary points and inflection Problem set A

Find the values of x for which:

1. (a) \(f(x)=x^3-9x^2+7x-2\) is concave-down;
2. (b) \(f(x)=x^3+6x^2-5x+1\) is concave-up.

Find the coordinates of the stationary points of \(y=x^3-9x^2+15x+4\), and classify each as a local maximum or a local minimum.

Find the coordinates of the stationary points of \(y=x^4-8x^2+3\), and classify each as a local maximum or a local minimum.

Find the coordinates of the stationary points of \(y=8x^{1/2}-3x+2\), and classify each as a local maximum or a local minimum.

Find the coordinates of the stationary points of \(y=e^x-4x\), 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 \(y=5\ln x-2x\), and classify each as a local maximum or a local minimum. Give coordinates to 3 significant figures.

\(f(x)=2x^3+kx^2-5x+4\) and \(f''(3)=20\). Find the value of k.

\(f(x)=x^3+ax^2+bx+7\), with \(f''(-2)=-18\) and \(f'(3)=14\). Find the values of a and b.

\(y=xe^{3x}\).

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

A company makes x hundred phones a week. Its weekly profit is \(P=-x^3+9x^2-15x\) thousand dollars, for \(0\le x\le 7\). 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 total sales of a new game, S thousand copies, t months after its launch are modelled by \(S=-t^3+12t^2+10t\) for \(0\le t\le 12\).

1. (a) Find \(S''\).
2. (b) Find the value of t at which \(S''=0\), and show that the concavity changes there.
3. (c) Explain why the sales are growing 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 \(y=3x^4+4x^3\). Classify each one. If the second derivative test does not decide, say what you do instead.