How do you find the max or minimum of #f(x)=-20x+5x^2+9#?

1 Answer
Mar 16, 2017

See below.

Explanation:

Because the function is #f(x)=5x^2-20x+9#, and the coefficient of the #x^2# term is positive, it will achieve a minimum at its vertex. To find the minimum, we need to express the parabola in vertex form.

#f(x)=5x^2-20x+9#
#f(x)=5(x^2-4x)+9#
To complete the square in the parenthesis, we need to add a #4# into the expression, but because there is a #5# on the outside, we actually add #20#. Now, because we added a #20# into the function, we need to subtract a #20# as well.

#f(x)=5(x^2-4x+4)+9-20#
#f(x)=5(x-2)^2-11#
Thus, the vertex is #(2, -11)#, and is the minimum.

graph{5x^2-20x+9 [-40, 40, -20.84, 20.84]}