Calculus Homework (derivatives)

Question

I was able to answer a I just can't figure out how to do b or c.

 

Let f(x)= x³e^-9x

 

a) Find the derivative of f ' (x)

 

I got this answer: -9e^-9xx³+3e^-9xx²

 

 

b) Find the critical numbers for f.

 

c) Determine where f is increasing or decreasing.

Answer 1

Your derivative is correct.

 

(b) Critical points are those points on the graph of f(x) where f'(x) is zero. So set the first derivative equal to 0 and solve for x:

 

- 9 e^-9x x³ + 3 e^-9x x² = 0

 

Since e^-9x can never be equal to zero, the only way this equation would be true is if:

 

- 9 x³ + 3 x² = 0

x² (3 - 9x) = 0

x = 0 or x = 1/3

 

So these are the critical numbers.

 

(c) We divide the entire number line in three parts based on the two critical points:

 

-inf < x < 0

0 < x < 1/3

1.3 < x < inf

 

inf stands for infinity. 

 

Now just pick a number from each of these intervals and plug in to the first derivative and just note the sign of the derivative which says whether the function is decreasing or increasing:

 

-inf < x < 0     sign of f'(x) = +, thus function is increasing
0 < x < 1/3     sign of f'(x) = +, thus function is increasing
1.3 < x < inf    sign of f'(x) = -, thus function is decreasing