Let f(x) be a function defined on the interval (0,) . We want to compare f(x) to the function x^(0.9)=1/x^(0.9) .

For each of the following scenarios, choose the true conclusion that can be drawn from the comparison.

Suggestion: Pay special attention to the integration area!

1) Knowing that 0f(x)1/x^(0.9) for all x(0,1] , we ...

(a) concludes that 10f(x)dx must converge.

(b) concludes that 10f(x)dx must diverge.

(c) cannot determine whether 10f(x)dx converges or diverges from this comparison.

2) Knowing that 01/x^(0.9)f(x) for all x(0,1] , we ...

(a) cannot determine whether 10f(x)dx converges or diverges from this comparison.

(b) conclude that 10f(x)dx must converge.

(c) concludes that 10f(x)dx must diverge.

3) Knowing that 0f(x)1/x^(0.9) for all x[1,) , we ...

(a) concludes that 1f(x)dx must converge.

(b) concludes that 1f(x)dx must diverge.

(c) cannot determine whether 1f(x)dx converges or diverges from this comparison.

4) Knowing that 01/x^(0.9)f(x) for all x[1,) , we ...

(a) concludes that 1f(x)dx must diverge.

(b) concludes that 1f(x)dx must converge.

(c) cannot determine whether 1f(x)dx converges or diverges from this comparison.

5) Knowing that 0f(x)1/x^(0.9) for all x(0,) , we ...

(a) concludes that 0f(x)dx must converge.

(b) cannot determine whether 0f(x)dx converges or diverges from this comparison.

(c) concludes that 0f(x)dx must diverge.

6) Knowing that 01/x^(0.9)f(x) for all x(0,) , we ...

(a) concludes that 0f(x)dx must diverge.

(b) concludes that 0f(x)dx must converge.

(c) cannot determine whether 0f(x)dx converges or diverges from this comparison.