Question

Assertion :STATEMENT-1: If f(x) is continuous on $[a,b]$, then there exists a point $c\in (a,b)$ such that ${\int }_a^{b}f\left(x\right)dx=f\left(c\right)(b-a).$ Reason: STATEMENT-2: For $a<b$, if m and M are, respectively, the smallest and greatest values of $f\left(x\right)$ on $[a,b]$,

then $m(b-a)\le {\int }_a^{b}f\left(x\right)dx\le (b-a)M.$

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1.
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1.
• C. STATEMENT-1 is True, STATEMENT-2 is False.
• D. STATEMENT-1 is False, STATEMENT-2 is True.

Answer 1

The correct option is A STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1.

For $a<b$, if m and M are the smallest and greatest values of $f\left(x\right)$ on $[a,b]$, respectively, then
$m(b-a)\le {\int }_a^{b}f\left(x\right)dx\le (b-a)M$
or $m\le \frac{1}{(b-a)}{\int }_a^{b}f\left(x\right)dx\le M$
Since $f\left(x\right)$ is continuous on $[a,b]$, it takes on all intermediate values between m and M.
Therefore,
for some values $f\left(c\right),f(a\le f(c)\le b)$, we will
have $\frac{1}{(b-a)}{\int }_a^{b}f\left(x\right)dx=f\left(c\right)or{\int }_a^{b}f\left(x\right)dx=f\left(c\right)(b-a)$
Hence, both the statements are true and statement 2 is a correct explanation of statement 1.