Further problems 2

Exercise 1
Q1

Consider the graph of the function ff (shown below).

  1. Indicate the local extrema and saddle points.
  2. Draw the graph of f′′f^{\prime\prime}.
  3. Estimate the value f′(0)f^\prime(0) and f′′(0)f^{\prime\prime}(0).
Q2

Determine f(4)(1)f^{(4)}(1)

  1. f(x)=(x−2)2(x+2)2f(x)=(x-2)^2(x+2)^2
  2. f(x)=3cos⁡(x)−3x5f(x)=3\cos(x)-3x^5
  3. f(x)=x2/3f(x)=x^{2/3}
  4. f(x)=2x2f(x)=\frac{2}{x^2}
Q3

The function f(x)=x(x−1)3+2f(x)=x(x-1)^3+2 has stationary points at x=1x=1. Verify this statement, and classify the stationary point.

Q4

Determine the stationary points and classify them.

  1. f(x)=x(x−1)f(x)=\sqrt{x}(x-1)
  2. g(x)=1x+3xg(x)=\frac{1}{x}+3x
Q5

A polynomial ff of degree 33 has the following properties: f(0)=1,f′(0)=2,f′′(0)=3f(0)=1, f^\prime(0)=2, f^{\prime\prime}(0)=3, and f′′′(0)=4f^{\prime\prime\prime}(0)=4. Determine the function equation of ff.

Q6

Find a function whose third derivative is everywhere 22.

Solution