Find the volume of the solid obtained by rotating the region

Question

Solve the below:

The graph of the derivative of a function f is shown below.

(a) Over what intervals is f(x) increasing? decreasing?Why?

(b) At what x values does f(x) have a local maximum? Why?

(c) At what x values does f(x) have a local minimum? Why?

(d) Sketch a possible graph of f(x).

Find the volume of the solid obtained by rotating the region bounded by xy = 4 and y=(x-3)⁴about the x-axis.

Let f be a function that is defined and twice differentiable for all values of x and has the following properties:

f(1) = 3 lim _x→0⁺ f(x)=∞

f'(x)<0 when x>0

f'(x)>0 when x><0

lim _x→0⁺ f(x)=2

Let g(x)= f(1/x)Find the following

(a) g(1)
(b) lim _{x→0^{+ g}}(x)