Originally Posted by

**lysserloo** **Problem:** Given $\displaystyle f(x) = x^3 - ax^2 + 3x + b$,

(a) Determine conditions on a and b so that f(x) has exactly one critical point.

(b) Could f(x) have no critical points? If yes, determine the necessary

conditions on a and b. If no, explain why not.

(c) Determine conditions on a and b so that f(x) has an inflection point.

**What I've Done:**

Could anybody help me with this? All I've done so far is try to take the derivative, and I got:

$\displaystyle f\prime(x) = 3x^2 -2ax + 3$

Then I set that to 0, and realized that this would give me more than one critical point, so I didn't bother to try and solve for it.

Could somebody walk me through HOW to solve each of the parts of the question, and explain what you do?

Thank you SO much!