What is the minimum value of f(x)=3x^2-6x+12f(x)=3x26x+12?

1 Answer
Dec 5, 2015

99

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]}