Chapter 1, Problem 439RE

True or False. Justify your answer with a proof or a counterexample. Assume all functions f and g are continuous aver their domains.

439. If $f\left(x\right)>0,f^{\prime }\left(x\right)>0$ for all x, then the right-hand rule underestimates the integral ${\displaystyle {\int }_a^{b}f\left(x\right)}$ . Use a graph to justify your answer.

To determine

To check: whether the given statement, “If $f\left(x\right)>0,f\text{'}(x)>0$ for all x , then the right hand rule underestimates the integral ${\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}$ ” is true or false.

Answer to Problem 439RE

The given statement is false.

Explanation of Solution

Given information: It is given that $f\left(x\right)>0$ and $f\text{'}(x)>0$ , for all x .

Concept used:

  $f^{\prime }\left(x\right)>0\Rightarrow f\left(x\right)\text{ is increasing}\text{.}$

Calculation:

$f\left(x\right)$ is continuous over $\left[a,b\right]$ .

$f\left(x\right)>0$ , for all $x\in \left[a,b\right]$ .

  $f^{\prime }\left(x\right)>0\Rightarrow f\left(x\right)\text{ is increasing}\text{.}$

  $R_n={\displaystyle \sum _1^{n}f(x_n)}\Delta x$

If function $f\left(x\right)$ is increasing, then right hand rule overestimates the value of the integral.

Conclusion:

Hence, the given statement is false.