Question

H(x) = −(x − 5)2 + 3

g(x) = 4 cos(2x − π) − 2
f(x)

x y
0 −5
1 0
2 3
3 4
4 3
5 0
6 −5
Which function has the largest maximum?

Answers:
 f(x)
 g(x)


 h(x)  
All three functions have the same maximum value.

Answer 1

The correct option is 1. The function f(x) has the largest maximum.

Explanation:

The vertex form of a parabola is

`y=a(x-h)^2+k`

Where, (h,k) is vertex.

The given functions is

`h(x)=-(x-5)^2+3`

Here, a=-1, h=-5 and k=3. Since the value of a is negative, therefore it is an downward parabola and vertex is the point of maxima.

Thus the maximum value of the function h(x) is 3.

`g(x)=4\cos (2x-\pi)-2`

The value of cosine function lies between -1 to 1.

`-1\leq \cos (2x-\pi)\leq 1`

Multiply 4 on each side.

`-4\leq 4\cos (2x-\pi)\leq 4`

Subtract 2 from each side.

`-4-2\leq 4\cos (2x-\pi)-2\leq 4-2`

`-6\leq 4\cos (2x-\pi)-2\leq 2`

Therefore the maximum value of the function g(x) is 2.

From the given table it is clear that the maximum value of the function f(x) is 4 at x=3.

Since the function f(x) has the largest maximum, therefore the correct option is 1.

Answer 2

Max(f(x)) = f(3) = 4

Max(g(x)) = g(π/2) = 2

Max(h(x)) = h(5) = 3

Of these values, f(3) = 4 is the largest.


The function with the largest maximum is f(x).