Question

Let $f\left(x\right)=x^2−2.$ if ${}^6{∫}_{3}f\left(x\right)dx=3f\left(c\right)$ for some $c∈(3,6)$, then the value of $c$ is equal to

• A. $\sqrt{12}$
• B. $\sqrt{21}$
• C. $\sqrt{19}$
• D. $\sqrt{17}$
• E. $\sqrt{13}$

Answer 1

Answer: (B)

Given,
$f(x)=x^2−2…(i)$
Now, $\underset{3}{\overset{6}{∫}}f(x)dx=3f(c)$
$⇒\underset{3}{\overset{6}{∫}}\left(x^2−2\right)dx=3f(c)$ [from Eq (i)]
$⇒{\left[\frac{x^3}{3}−2x\right]}_3^6=3f(c)$
$⇒[72−12−9+6]=3f(c)$
$⇒78−21=3f(c)$
$⇒3f(c)=57$
$⇒f(c)=19$
$⇒c^2−2=19$ [from Eq. (i)]
$⇒c^2=21$
$⇒c=\sqrt{21}$ which lies between $(3,6)$