Both the function f(x) and g(x) have same values.
What is vertex?
The vertex of a quadratic is the point where the graph intersects with the axis of symmetry (the line which divides the parabola in half). The vertex of a quadratic will be either a maximum or minimum.
Formula for vertex
[tex]x = \frac{-b}{2a}[/tex]
where,
b is the coefficient of x
and , a is the coefficient of [tex]x^{2}[/tex].
for, y substitute the value of x in f(x)
⇒ (x, f(x)) i.e. (x, y) will be the vertex
Maximum or minimum value
The minimum or maximum value of quadratic is given by the value of the function at the the vertex.
According to the given question
we have
[tex]f(x) = -x^{2} +2x+4[/tex]
and
[tex]g(x)= -(x-5)^{2} +5[/tex]
For the vertex or the function [tex]f(x)=- x^{2} +2x+4[/tex]
[tex]x =\frac{-2}{-2}[/tex] = 1
and, at x = 1
f(1) = -1+ 2+4= 5
⇒ vertex of f(x) is (1, 5)
As we know, the maximum value of y is given by y coordinate so, the maximum value of f(x) is 5 at x=1.
Similarly,
For g(x) = [tex]-(x - 5)^{2}+5 = -x^{2} -25+10x +5 =-x^{2} + 10x -20[/tex]
⇒ g(x) = [tex]-x^{2} +10x -20[/tex]
[tex]x = \frac{-10}{-2}[/tex] = 5
g(5) = -25 + 50 -20
g(5) = 5
⇒ vertex of g(x) is (5,5)
⇒ Maximum value of g(x) is 5 at x =5.
Hence, both the function have same maximum value.
Learn more about the maximum and minimum values here:
https://brainly.com/question/14316282
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