Find the minimum value of the function g (x)=x^2-4x-10

Question

Find the minimum value of the function

g(x)=x^2-4x-10

Answer 1

Calculus Solution

 

The minimum or maximum will occur when the slope = 0 (assuming there are local minima/maxima)

g(x)=x^2-4x-10

g'(x)=2x-4

 

when g'(x) = 0 there is either a relative minimum or maximum

 

2x - 4 = 0 --> x=2

 

So at x=2 there  is a minimum [the value of the second derivative at x=2 is positive, so it is a minimal value.]. Substituting x=2 into g(x) gives us the minimum value:

 

g(2) = 2^2 - 4x2 - 10 = 4 - 8 - 10 = -14

 

Geometry method

1. the center of a parabola is at x=-b/2a therefore x = -(-4)/(2*1) = 2

2. plug x=2 into the formula to find out that y = -14.

3. since the sign of "a" is positive, the parabola opens upward. therefore at x=2, it is the minimum.