which of the following intervals is f(x) decreasing.

Question

If f(x) = x^6 + 12x^5 + 37, on which of the following intervals is f(x) decreasing.

I calculate

f'(x)=6x^5+60x^4

f'(x)=0

6x^5+60x^4=0 solve for x

x=-10,0

test points

I got f(x) decreasing (-infinite,0)

But this is wrong answer.

Could you explain...thank you.

Answer 1

Your critical points are correct. So set up a number line with the critical points on them like this:

Now, pick test points in each region. -11 could be a test point in region A. -1 could be a test point in Region B. And +1 could be a test point in Region C like this:

Plug in the test point into f'(x) and you get:

f'(-11) = -87846 (meaning the slope of the tangent line is negative in this interval which says the function is decreasing from negative infinity to -10)

f'(-1) = 54 (meaning the slope of the tangent line is positive in this interval which says the function is increasing from -10 to 0)

f'(1) = 66 (meaning the slope of the tangent line is positive in this interval which says the function is increasing from 0 to positive infinity)

So the function is decreasing only from negative infinity to -10