a. To find the number of roots of h(x) between x = 1 and x = 5, we look for sign changes in the values of h(x) at the endpoints and within the interval. We can see that h(1) = 1 and h(5) = 3. Since there is a sign change between these two values (h(1) is positive and h(5) is positive), there must be at least one root within the interval. Therefore, h(x) has at least 1 root between x = 1 and x = 5.

b. To find the number of roots of h(x) between x = 1 and x = 7, we look for sign changes in the values of h(x) at the endpoints and within the interval. We can see that h(1) = 1 and h(7) = -1. Since there is a sign change between these two values (h(1) is positive and h(7) is negative), there must be at least one root within the interval. Therefore, h(x) has at least 1 root between x = 1 and x = 7.

c. To find the number of places where h(x) = 3.4 between x = 1 and x = 7, we need to determine if there is a value of x within the interval where h(x) takes on the value 3.4. From the given values, we can see that there is no value of x between 1 and 7 where h(x) equals 3.4. Therefore, h(x) does not equal 3.4 between x = 1 and x = 7.

d. To find the number of places where h(x) = -0.5 between x = 1 and x = 7, we need to determine if there is a value of x within the interval where h(x) takes on the value -0.5. From the given values, we can see that there is no value of x between 1 and 7 where h(x) equals -0.5. Therefore, h(x) does not equal -0.5 between x = 1 and x = 7.

e. To determine if it is possible for h(x) to equal π for some value(s) of x between 5 and 6, we need to examine the given values of h(x) within this interval. From the given values, we can see that there is no value of x between 5 and 6 where h(x) equals π. Therefore, it is not possible for h(x) to equal π for any value of x between 5 and 6.

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