: intervals of increase and decrease for f(x)=(x-3)(x^2-6x-3)

Answer 1

Increaseing and decreasing intervals end at extrema (maximum or minimum) or at infinity.

It would make it easy to first find f'(x), equate it to zero to find the values of x at which the extrema occur.

So
f(x)=(x-3)(x²-6x-3)
=x³-9x²+15x+9
f'(x)=3x²-18x+15
Set f'(x)=0 and solve for x to get x=5 or x=1.

The leading term of f(x) is x³, and its coefficient is positive, we know that f(x)->∞ as x->&infin, and similarly, f(x)->-∞ as x-> -∞.

All that is left to do is to figure out the intervals, and find if they are increasing or decreasing.