# Suppose that f(x)= ln((9x^2)+3) (A) Find all critical values of f, compute their average.   (B) Use interval notation to indicate where f(x) is increasing. (C) Use interval notation to indicate where f(x) is decreasing.(D) Find the x-coordinates of all local maxima of x, compute their average. (E) Find the x-coordinates of all local minima of f, compute their average. (F) Use interval notation to indicate where f(x) is concave up.(G) Use interval notation to indicate where f(x) is concave down.

A) You need to solve the equation f'(x) = 0 to find critical values of the function such that:

`f'(x) = ((9x^2+3)')/(9x^2+3)`

`f'(x) = (18x)/(9x^2+3)`

You need to solve the equation f'(x) = 0, hence since `(9x^2+3)!=0` , then `18x = 0 =gt x = 0.`

Hence, the critical value...

A) You need to solve the equation f'(x) = 0 to find critical values of the function such that:

`f'(x) = ((9x^2+3)')/(9x^2+3)`

`f'(x) = (18x)/(9x^2+3)`

You need to solve the equation f'(x) = 0, hence since `(9x^2+3)!=0` , then `18x = 0 =gt x = 0.`

Hence, the critical value of the function is x=0.

B) The function is increasing over `(0,oo).`

C) The function only increases over `(0,oo).`

D) You need to remember that the x coordinate of local maximum is the critical value of function, hence x = 0.

E) Since the function has no minimum, then there is no x coordinate for local minimum.

F) Since the function increases over`(0,oo),` hence the function is concave up over `(0,oo).`

G) Since the function only increases over `(0,oo)` ,hence the function is not concave down over any interval.

Approved by eNotes Editorial Team