Question

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

Answer 1

Critical point x=`2/3`
Minima `-1/3`
No maxima, no inflection point.

Explanation:

Find f'(x) = 6x-4. Equate it to 0 to get `x=2/3`. This is the only critical point.

Now find f"(x) =6, which is a positive number (irrespective of any x) . Hence, there is a minima at x=2/3. The minima would be `3(2/3)^2 -4(2/3) +1`

= `-1/3`

Maxima does not exist.

Since f"(x) `!=` 0, there is no inflection point.