Question

Assertion :For the function $f\left(x\right)=x^2+3x+2$, $LMVT$ is applicable in $[1,2]$ and the value of $c$ is $\frac{3}{2}$. Reason: If $LMVT$ is known to be applicable for any quadratic polynomial in $[a,b]$, then $c$ of $LMVT$ is $\frac{(a+b)}{2}$

• 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

$f\left(x\right)=x^2+3x+2$
$f\left(1\right)=6$, $f\left(2\right)=12$
Also, $f^{\prime }\left(x\right)=2x+3$
Cleary f(x) is continuous and differentiable in [1,2], so by LMVT, there exists $c\in (1,2)$ such that
$f^{\prime }\left(c\right)=\frac{f\left(2\right)-f\left(1\right)}{2-1}$
$2c+3=6$
$\Rightarrow c=\frac{3}{2}$
Hence, assertion is true.
If LMVT is applicable for any quadratic polynomial in [a,b], then value of c is $\frac{a+b}{2}$