Determine f such that f ′′(x) = 6x − 6 and 2 is both a critical number and an x intercept of f.

Question

Determine f such that f ′′(x) = 6x − 6 and 2 is both a critical number and an x intercept of f.

Answer 1

Nice problem. First begin by finding the anti-derivative twice keeping in mind that we generate unknown constants each time we determine an indefinite integrarl

if f''(x) = 6x - 6

then f'(x) = 3x^2 - 6x + A

f(x) = x^3 - 3x^2 + Ax + B (A and B are the unknown constants.

If 2 is a critical number, this means that the derivative of the function when x = 2 is 0 so

0 = 3(2)^2 - 6(2) + A

A = -6

so f(x) now is x^3 - 3x^2 - 6x + B

Now if x=2 means it's also x-intercept, then y = 0, or the point (2,0) is on the function Plug this point in and resolve for B. I cam up with

f(x) = x^3 - 3x^2 - 6x + 16