Question

Assertion :

Let $f\left(x\right)$ is continuous and positive for $x\in [a,b],g\left(x\right)$ is continuous for $x\in [a,b]$ and ${\int }_a^b\left|g\right(x\left)\right|dx>\left|{\int }_a^{b}g\left(x\right)dx\right|$

The value of ${\int }_a^{b}f\left(x\right)g\left(x\right)dx$ can be zero.

Reason: Equation $g\left(x\right)=0$ has at least one root for $x\in (a,b)$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is Not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

${\int }_a^{b}lg\left(x\right)dx>\left[{\int }_a^{b}g\left(x\right)dx\right]\Rightarrow x^e[a,b]$

$\Rightarrow g\left(x\right)⟶$ Continuious for $x\in [a,b]$

$\Rightarrow {\int }_a^{b}f\left(x\right)g\left(x\right)dx⟶$ Can be zero

If $f\left(x\right)$ Or $g\left(x\right)=0$

$\Rightarrow g\left(x\right)$

Should have one root between $(1,0)$

Hence Both Assertion and Reason are correct and

Reason is the correct explanation for Assertion