The table below gives selected values of a continuous function $f$. If $f$ is increasing over the closed interval $[0,3]$ which of the following could be the value of $\int^{3}_{0} f(x)dx $?
The table given is:
x: 0 .5 1 1.5 2 2.5 3
f(x): 0 4 10 18 28 40 54
Narrowing down the answers, I determined that the integral must be less than $77$, since that is the maximum sum of the Reimann sum, if the function was not always increasing.
However, the last two choices are $50$ or $62$. Why is $50$ an impossibility?
$$\int^{3}_{0} f(x)dx=\int^{0.5}_{0} f(x)dx+\int^{3}_{0.5} f(x)dx$$ You can easily obtain by summing : $$\int^{3}_{0.5} f(x)dx\geq 50$$ But $f(0.5)=4$ and $f$ is continuous, so $\exists a \in (0,0.5) $ such that on $[a,0.5],f>0$. Furthermore $f\geq0$ on $[0,0.5]$ since $f$ is increasing so : $$\int^{0.5}_{0} f(x)dx>0$$ Finally : $$\int^{3}_{0} f(x)dx>50$$