For the polynomial, P(x)=x^4-4x^3+10, over what intervals is this function incr. and decr.?

Question

Identify and describe any local extrema.

Over what intervals is this function concave up or down?

Answer 1

 P(x)=x⁴-4x³+10, P'(x)=4x³-12x²=4x²(x-3)=0 at extrema so x=0 and 3.

P''(x)=12x²-24x=12x(x-2), P''(x)=0 at x=0 and P''(x)=0 at x=0.

There is a saddle-point at x=0 and a minimum at x=3. The slope is decreasing (concave down) while x<0, levels out close to x=0, continues to decrease until x=3.

For x<3 P is decreasing (concave down) towards the minimum at x=3, and for x>3 it is increasing (concave up). When x slightly less than zero and slightly more than zero it is neither increasing or decreasing.