Let f(x) = x^2 − 9x − 52.

Question

a) Find the interval(s) where f(x)  is increasing. (Enter your answer using interval notation.)

b) Find the interval(s) where f(x) is decreasing. (Enter your answer using interval notation.)

c)Find the x-value(s) of all local maxima of f(x).

 (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

 

(d)

Find the x-value(s) of all local minima of f(x).

 (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

 

Answer 1

The first step is to find the derivative.

f(x)=x²-9x-52

f'(x) = 2x-9 Set the derivative equal to zero to find the critical value

2x-9=0

2x=9

x=9/2=4.5 Now will need to pick test points to see what the signs are for each region.

at x=0 2(0)-9 is negative, the function is decreasing here

at x=5 2(5)-9 is positive, the function is increasing here

Since the function goes from decreasing to increasing, x=9/2 must be a local min. There is no local max, there is only one critical value.

a) Find the interval(s) where f(x)  is increasing. (Enter your answer using interval notation.)

(9/2,∞)

b) Find the interval(s) where f(x) is decreasing. (Enter your answer using interval notation.)

(-∞,9/2)

c)Find the x-value(s) of all local maxima of f(x).

 (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

DNE

d) Find the x-value(s) of all local minima of f(x).

 (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x=9/2