How do you find all critical point and determine the min, max and inflection given #f(x)=3x^2-4x+1#?

Maxima does not exist.

To find all critical points, we first find the derivative of the function, $f'(x)$. Then, we set $f'(x) = 0$ and solve for $x$. The critical points are the $x$ values where the derivative is zero or undefined.

Next, to determine whether each critical point corresponds to a minimum, maximum, or inflection point, we examine the sign of the second derivative, $f''(x)$, at each critical point. If $f''(x) > 0$, the function has a local minimum at that critical point. If $f''(x) < 0$, the function has a local maximum at that critical point. If $f''(x) = 0$, the test is inconclusive, and we need further investigation.

To find inflection points, we locate the points where the concavity changes, which occurs where $f''(x) = 0$ or is undefined.

Let's follow these steps:

1. Find $f'(x)$ by differentiating $f(x) = 3x^2 - 4x + 1$.
2. Set $f'(x) = 0$ and solve for $x$ to find critical points.
3. Compute $f''(x)$ by differentiating $f'(x)$.
4. Evaluate $f''(x)$ at each critical point to determine the nature of the critical points.
5. Identify any inflection points by finding where $f''(x) = 0$ or is undefined.

Following these steps will help us find the critical points, as well as determine whether they correspond to minima, maxima, or inflection points.