Determine the location of each local extremum of the function.

So basically, when a function is at it's absolute highest value, the slope is at zero. How do you find the slope you ask? Derivatives!

f(x)=x^3+9x^2+15x+1
f'(x)=3x^2+18x+15

Now, because the slope is equal to zero when a function is a min/max, we get f'(x)=0

0=3x^2+18x+15
0=x^2+6x+5
0=(x+5)(x+1)

The zeroes are at x=-1 and x=-5

Now plug these values into f(x)
f(-1)=(-1)^3+9(-1)^2+15(-1)+1=-6
f(-5)=(-5)^3+9(-5)^2+15(-5)+1=26

max: (-5,26)
min: (-1,-6)