Answer:
A)<em> g(x) </em>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:
![f(x) =-x^{2}+4x-5](https://tex.z-dn.net/?f=f%28x%29%20%3D-x%5E%7B2%7D%2B4x-5)
Let us convert it to vertex form to find its vertex.
Taking - sign common:
![f(x) =-(x^{2}-4x+5)](https://tex.z-dn.net/?f=f%28x%29%20%3D-%28x%5E%7B2%7D-4x%2B5%29)
Now, let us try to make it a whole square,
Writing 5 as 4+1:
![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](https://tex.z-dn.net/?f=f%28x%29%20%3D-%28x%5E%7B2%7D-4x%2B4%2B1%29%5C%5C%5CRightarrow%20f%28x%29%20%3D-%28%28x%5E%7B2%7D-2%20%5Ctimes%202%5Ctimes%20x%2B2%5E2%29%2B1%29%5C%5C%5CRightarrow%20f%28x%29%20%3D-%28%28x-2%29%5E%7B2%7D%2B1%29%5C%5C%5CRightarrow%20f%28x%29%20%3D-%28x-2%29%5E%7B2%7D-1)
Please refer to attached graph of f(x).
We know that, vertex form of a parabola is given as:
![f (x) = a(x - h)^2 + k](https://tex.z-dn.net/?f=f%20%28x%29%20%3D%20a%28x%20-%20h%29%5E2%20%2B%20k)
Comparing the equations we get:
a = -1 (Negative value of <em>a</em> 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)<em> g(x) </em>has a greater absolute maximum.