If -1 < h < 0, which of the following expressions has the least value?

Rationale

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

In this interval, the value of h is negative, and raising a negative number to a power results in a negative number, while higher powers of h will yield even more negative values. Thus, among the given expressions, h^3 will have the least value as it decreases more rapidly than the others.

A (h^2 - 2h + 1) YOUR ANSWER

This expression can be rewritten as (h - 1)^2, which is always non-negative for any real h. Since h is between -1 and 0, this expression will yield values between 0 and 1.

B (h^2 - h) YOUR ANSWER

This expression represents a quadratic function that opens upwards. It takes values between 0 and 1 in the interval -1 < h < 0, as the maximum occurs at h = 0 and the minimum occurs at h = -1, where it equals 0.

C (h) YOUR ANSWER

Since h is negative within the specified range, this expression will take values between -1 and 0. However, it does not decrease as sharply as the cubic expression h^3.

D (h^2) YOUR ANSWER

This expression will always be positive for any non-zero h and will range from 0 to 1 within the interval -1 < h < 0. Thus, it cannot be less than h^3.

Conclusion

In the interval -1 < h < 0, the expression h^3 yields the least value, as it becomes more negative compared to the other expressions. While all other choices either remain non-negative or less negative than h^3, the cubic function's rapid decrease ensures it achieves the lowest value within the given constraints.

If -1 < h < 0, which of the following expressions has the least value?

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

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