Question

What is the minimum value of `f(x)=3x^2-6x+12`?

Answer 1

`9`

Explanation:

Relative minimum and maximum points may be found by setting the derivative to zero.
In this case,
`f'(x)=0 iff6x-6=0`
`iff x=1`

The corresponding function value at 1 is `f(1)=9`.

Hence the point `(1,9)` is a relative extreme point.

Since the second derivative is positive when x = 1, `f''(1)=6>0`, it implies that x = 1 is a relative minimum.

Since the function f is a 2nd degree polynomial, its graph is a parabola and hence `f(x)=9` is also the absolute minimum of the function over `(-oo,oo)`.
The attached graph also verifies this point.

graph{3x^2-6x+12 [-16.23, 35.05, -0.7, 24.94]}