Question

Let $f \left(x\right)=x^{2}-2.$ if $^{6}\int_{3}f \left(x\right)dx=3f \left(c\right)$ for some $c \in (3, 6)$, 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,
$f(x)=x^{2}-2\,\,\,\,\,\dots(i)$
Now, $\int\limits_{3}^{6} f(x) d x=3 f(c)$
$\Rightarrow \, \int\limits_{3}^{6}\left(x^{2}-2\right) d x=3 f(c)$ [from Eq (i)]
$\Rightarrow \, \left[\frac{x^{3}}{3}-2 x\right]_{3}^{6}=3 f(c)$
$\Rightarrow \,[72-12-9+6]=3 f(c)$
$\Rightarrow \, 78-21=3 f(c)$
$\Rightarrow \, 3 f(c)=57 $
$\Rightarrow \, f(c)=19$
$\Rightarrow \, c^{2}-2=19$ [from Eq. (i)]
$\Rightarrow \, c^{2}=21$
$\Rightarrow \,c=\sqrt{21}$ which lies between $(3,6)$