Question

If $f\left(x\right)={\int }_1^x\frac{dt}{2+t^4}$, then

• A. $f\left(2\right)<\frac{1}{3}$
• B. $f\left(2\right)>\frac{1}{3}$
• C. $f\left(2\right)=\frac{1}{3}$
• D. $f\left(2\right)<1$

Answer 1

The correct option is A $f\left(2\right)<\frac{1}{3}$

$f^{{}^{\prime }}\left(x\right)=\frac{1}{2+x^4}$
By LMVT $f^{{}^{\prime }}\left(C\right)=\frac{f\left(2\right)-f\left(1\right)}{2-1}$ for some $cϵ(1,2)$
$\Rightarrow f\left(2\right)=\frac{1}{2+c^4}$ as $f\left(1\right)=0\Rightarrow 1<c<2\Rightarrow 3<2+c^4<18\Rightarrow f\left(2\right)<\frac{1}{3}$