Exercise 6

For an unknown function $f(x)$, the following information is known.

• $f$ is continuous on $[3,6]$;

• $f$ is either always increasing or always decreasing on $[3,6]$;

• $f$ has the same concavity throughout the interval $[3,6]$;

• As approximations to $\int_3^6 f(x) \, dx$, $L_4 = 7.23$, $R_4 = 6.75$, and $M_4 = 7.05$.

1. Is $f$ increasing or decreasing on $[3,6]$? What data tells you?

2. Is $f$ concave up or concave down on $[3,6]$? Why?

3. Determine the best possible estimate you can for $\int_3^6 f(x) \, dx$, based on the given information.