Find the minimum values of g(x) =5x^2 - 5x + 7

Given the parabola g(x) = 5x^2 - 5x +
7.

We need to find the minimum
value.

First, we need to find the first
derivative.

==> g'(x) = 10x -
5

Now we will calculate the critical values which is the derivative
zeros:

==> 10x - 5 =
0

==> 10x = 5

==> x =
1/2.

Since the sign of x^2 is positive, then we know that the
function has a minimum value at x= 1/2

Then the function has a
minimum value at x= 1/2

==> g(1/2) = 5(1/2)^2 - 5(1/2) +
7

                = 5/4 - 5/2 +
7

                 = ( 5 - 10 +
28)/4

                  =
23/4

Then, the minimum values is: f(1/2) =
23/4