I'm quite confused on this Calculus 1 question I have. I'm given that:
f(x) = 4x3 + 3x2 − 6x + 2
The goal is to find the intervals at which it increases and the intervals at which it decreases.
I found that the derivative of f(x) is 12x2+6x-6 and I set it equal to 0 to find the critical points. I found -1 and 1/2 to be the critical points. Now, I chose some numbers on each interval to determine whether it is increasing or decreasing (based on being positive of negative) and plugged it into the original equation.
If I put in some number less than -1 (like -10 for instance) it is positive. If I put in some number in between -1 and 1/2 (like 0 for instance) it is positive. If I put in a number greater than 1/2 (1 for instance) it is positive.
My confusion is I decided to graph it just to make sure and the graph shows that the interval has a negative slope from -1 to 1/2 but I don't understand why. I hope someone can help, thank you!
Important numbers if you want a TL;DR:
f(x) = 4x3 + 3x2 - 6x + 2
f'(x) = 12x2 + 6x - 6
Set f'(x) = 0 for critical points, which is when x = -1 and .5
Find intervals of increase and decrease.