Answer:
A) g(x) has a greater absolute maximum.
Step-by-step explanation:
Given graph of g(x) which is a Parabola
1. Opens downwards
2. The absolute maximum (vertex) is at around (3.5, 6)
i.e. value of absolute maximum is 6.
Another function:
[tex]f(x) =-x^{2}+4x-5[/tex]
Let us convert it to vertex form to find its vertex.
Taking - sign common:
[tex]f(x) =-(x^{2}-4x+5)[/tex]
Now, let us try to make it a whole square,
Writing 5 as 4+1:
[tex]f(x) =-(x^{2}-4x+4+1)\\\Rightarrow f(x) =-((x^{2}-2 \times 2\times x+2^2)+1)\\\Rightarrow f(x) =-((x-2)^{2}+1)\\\Rightarrow f(x) =-(x-2)^{2}-1[/tex]
Please refer to attached graph of f(x).
We know that, vertex form of a parabola is given as:
[tex]f (x) = a(x - h)^2 + k[/tex]
Comparing the equations we get:
a = -1 (Negative value of a means the parabola opens downwards)
h = 2, k = -1
Vertex of f(x) is at (2, -1) i.e. value of absolute maximum is -1
and
Vertex of g(x) is at (3.5, 6)
i.e. value of absolute maximum is 6.
Hence, correct answer is:
A) g(x) has a greater absolute maximum.
