Question

Assertion :Let f be a polynomial function of degree n.
STATEMENT-1: There exists a number $x\in [a,b]$ such that ${\int }_a^{x}f\left(t\right)dt={\int }_x^{b}f\left(t\right)dt$. Reason: STATEMENT-2: $f\left(x\right)$ is a continuous function.

• 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.

Let $g\left(x\right)={\int }_a^{x}f\left(t\right)dt-{\int }_x^{b}f\left(t\right)dt,\text{where}x\in [a,b]$
We have $g\left(a\right)=-{\int }_a^{b}f\left(t\right)dt$ and $g\left(b\right)={\int }_a^{b}f\left(t\right)dt$
$\therefore g\left(a\right)g\left(b\right)=-{\left({\int }_a^{b}f\right(t\left)dt\right)}^2\le 0$
Clearly, $g\left(x\right)$ is continuous in $[a,b]$ and $g\left(a\right)g\left(b\right)<0$.
It implies that $g\left(x\right)$ will become zero at least once in $[a,b]$.
Hence, ${\int }_a^{x}f\left(t\right)dt={\int }_x^{b}f\left(t\right)dt$ for at least one value of $x\in [a,b]$.
Hence, both the statements are true and statement 2 is a correct explanation of statement 1.