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Compare f(2) with 1/3 for f(x)=∫dt/(2+t⁴) | JEE

JEE Maths question with a full step-by-step solution.

Question

f(x)=\displaystyle\int_{1}^{x}\dfrac{dt}{2+t^{4}}

, then

f(2)<\dfrac13

correct

f(2)>\dfrac13

f(2)=\dfrac13

f(2)>1

Solution

Step 1: Differentiate by the Fundamental Theorem of Calculus:

f'(x)=\dfrac{1}{2+x^{4}}

, and note

f(1)=0

. Step 2: Apply the Lagrange Mean Value Theorem to

f

[1,2]

. There exists

c\in(1,2)

with

f'(c)=\dfrac{f(2)-f(1)}{2-1}=f(2)\quad\Rightarrow\quad f(2)=\dfrac{1}{2+c^{4}}.

Step 3: Bound

c^{4}

. Since

1<c<2

, we have

1<c^{4}<16

, hence

3<2+c^{4}<18

, giving

\dfrac{1}{18}<\dfrac{1}{2+c^{4}}<\dfrac{1}{3}.

Step 4: Therefore

f(2)=\dfrac{1}{2+c^{4}}<\dfrac13.

Correct answer: (1)

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