How do you find all points of inflection of the function f(x)=x^3-3x^2-x+7f(x)=x33x2x+7?

1 Answer
May 20, 2015

The inflection of a function means the changes in its direction. You **can either understand it as the change from increasing to decreasing (and vice-versa) on your f(x)f(x) or the vertex of the slope for your f'(x).

But how do we find inflection points? Using your function's second derivatives f''(x).

Just a quick reminder: your f'(x) measures the increase / decrease rate of f(x), while f''(x) indicates whether f(x) changes from convex to concave (or vice-versa).

Thus, we must equal the second derivative of the function to zero.

f'(x)=3x^2-6x-1,
f''(x)=6x-6.

Equaling it to zero, we have 6x=6, x=1.

For f(1)=1^3-3(1^2)-1+7=4

Thus, your inflection point is (1,4)

graph{x^3-3x^2-x+7 [-15.27, 16.33, -2.72, 13.08]}