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.

(-1,-0.5)

step-by-step explanation:

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."