Where is 3x raised to the 4th plus 4x raised to the 3rd -36x squared plus 1 increasing?

Answer 1

(3x)^4 +(4x)^3 -(36x)^2 +1 or 3x^4 + 4x^3 -36x^2 +1? Not sure from the way you said it.

In either case the method is the same. Use the power rule for differentiation, d/dx(x^n) = nx^(n-1), to take the derivative of the equation term by term, and then set this equal to zero and solve for x. You will get a cubic function when you do this, so you can try to factor it after you differentiate to simplify the problem.

Assuming the second case (It seems more likely) differentiating gives the following, which I have set equal to zero:

12x^3 + 12x^2 - 72x=0

x(12x^2 +12x-72)=0 Factor out an x

x(12)(x-2)(x+3)=0 Factor some more

This gives horizontal tangents at x=-3,0,and 2. Now all you have to do is plug some easy numbers into the derivative from the intervals between these values. Negative derivative means decreasing, positive means increasing. Good luck!