Question

Which function has the smallest minimum y-value?

fx) = 2 sin (3x + π) − 2
g(x) = (x − 3)2 − 1
h(x) >
x y
−2 3
−1 −2
0 −5
1 −6
2 −5
3 −2
4 3

Answer 1

The minimum value of sine is negative one, so f(x) has minimum -2-2=4. The square function has minimum 0, so g(x) has minimum 0-1=-1.
h(x) has minimum -6 when x=1, so it has the smallest minimum.

Answer 2

The smallest minimum is attained by the function:

h(x)

Explanation:

We are asked to find which function has smallest minimum value:

We have:

• f(x)=2 sin (3x+π)-2

We know that the minimum value of sine function is -1 and the maximum value of sine function is 1.

So, when sine function will have minimum value -1 then the function f(x) also has minimum value as -4.

( since 2×(-1)-2=-2-2= -4 )

• g(x)=(x-3)^2-1

As this function is a quadratic function and we know that:

(x-3)^2≥0 for all x.

so,

(x-3)^2-1≥ -1.

Hence, the minimum value of g(x) is -1.

• Also we are given a set of values of function h(x) as:

h(x) =y

x y

−2 3

−1 −2

0 − 5

1 −6

2 −5

3 −2

4 3

Clearly from the table we could see that h(x) receives -6 as the minimum value.

Hence, the smallest minimum is attained by the function h(x).