Question

the graphs of three functions are given below: f (in blue), g (in green), and h (in red). these functions are continuous on (0, ∞). assume that the graphs continue in the same way as x goes to infinity (i.e. green stays on top, blue in the middle, red on the bottom). suppose ∫₁^∞ f(x) dx is convergent. which of the following do you know to be true by the comparison test? ∫₁^∞ g(x) dx diverges ∫₁^∞ h(x) dx diverges ∫₁^∞ g(x) dx converges ∫₁^∞ h(x) dx converges

Explanation:

From the graph and description, the order of the functions for \(x \geq 1\) is \(g(x) \geq f(x) \geq h(x) > 0\) (since they stay green on top, blue middle, red bottom and approach the x-axis). The Comparison Test states: if \(0 < A(x) \leq B(x)\) and \(\int_1^\infty B(x)dx\) converges, then \(\int_1^\infty A(x)dx\) converges; if \(0 < B(x) \leq A(x)\) and \(\int_1^\infty B(x)dx\) diverges, then \(\int_1^\infty A(x)dx\) diverges. Given \(\int_1^\infty f(x)dx\) converges and \(h(x) \leq f(x)\), \(\int_1^\infty h(x)dx\) must converge. We cannot conclude about \(g(x)\) because \(g(x) \geq f(x)\) (a larger function's integral could diverge even if the smaller one converges).

Answer:

\(\int_{1}^{\infty} h(x) dx\) converges