Verified Answer :

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:

The smallest minimum is attained by the function:

h(x)

Step-by-step 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).