Question 10

h^3 has the least value when -1 < h < 0.

Among the expressions provided, h^3 yields the least value in the interval where h is between -1 and 0.

A) h^2 - 2h + 1

This expression can be rewritten as (h - 1)^2, which is a perfect square. Since h is between -1 and 0, (h - 1) will be negative, making (h - 1)^2 positive. Therefore, this expression cannot attain the least value in the specified range.

B) h^2 - h

This expression can be factored as h(h - 1). Given that h is negative, both h and (h - 1) are negative, resulting in a positive product. Thus, this expression also does not have the least value in the interval of interest.

C) h

The expression h itself is negative in the interval -1 < h < 0. While it is less than the other expressions also evaluated, it is not the least value when compared to h^3, which is a more negative value in this range.

D) h^2

This expression, being a square, is always non-negative. Specifically, h^2 will yield values between 0 and 1 for h within the range of -1 to 0, and thus it cannot be the least value when compared to negative values of h^3.

E) h^3

The expression h^3 yields negative values that are more negative than h, h^2, or any other expression in the given interval. As h approaches 0 from the left, h^3 approaches 0 but remains negative, making it the expression with the least value in the specified range.

Conclusion

In the interval -1 < h < 0, h^3 is the least value compared to the other expressions due to its negative output. All other options either yield non-negative values or are less negative than h^3, confirming that h^3 is indeed the correct answer for having the least value.