9. Let f be a continuous function such that f(3) = 4 and f(5)=2. Let h(x)= f(x) + 2x. Explain why h(t)=11 for some t on the interval 3

Since f(x) is continuous, so is h(x).

Also, since f (3) = 4 and f (5) = 2, we have

h (3) = f (3) + 2•3 = 4 + 6 = 10

h (5) = f (5) + 2•5 = 2 + 10 = 12

It follows from the intermediate value theorem that there is some t between 3 and 5 such that h(t) is between 10 and 12, and namely there is some t for which h(t) = 11.

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