SOLUTION: For what values of x is the function f(x)= 5+15x+6x^2-x^3 decreasing?

Algebra ->  Test -> SOLUTION: For what values of x is the function f(x)= 5+15x+6x^2-x^3 decreasing?      Log On


   

Question 262269: For what values of x is the function f(x)= 5+15x+6x^2-x^3 decreasing?
Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
here is the problem:
(i) f%28x%29=+5%2B15x%2B6x%5E2-x%5E3
to determine increasing and decreasing, take a second derivative as
(ii) f%27%28x%29+=+15+%2B+12x+-+3x%5E2
set = 0 and find x. We get
(iii) f%27%28x%29+=+15+%2B+12x+-+3x%5E2+=+0+
divide by -3 to get
(iv) x%5E2+-+4x+-+5+=+0
(v) %28x%2B1%29%28x-5%29+=+0
so, x = -1 and x = 5.
these are critical values.
From (ii) take a second derivative as
(vi) f''(x) = 12+-+6x+=+0
and solving for x, we get
x+=+2
this is also a critical value.
pick a number less than and greater than 2 in to f'' and we get
x= 0 - - > increasing ; x = 4 - - > decreasing.
--
so the intervals of increasing and decreasing are
increasing (-00, 2)
decreasing (2, +00)
here is the graph to help.
+graph%28+300%2C+200%2C+-10%2C+10%2C+-20%2C+100%2C+5%2B15x%2B6x%5E2-x%5E3%29+