Let the function f:0,1→ℝ be defined by fx=4x4x+2. Then the value of f140+f240+f340+⋯+f3940-f12 is_________

JEE Advanced · Mathematics · 22. Functions

Let the function $f : [0, 1] \to ℝ$ be defined by $f (x) = \frac{4^{x}}{4^{x} + 2}$ . Then the value of $f (\frac{1}{40}) + f (\frac{2}{40}) + f (\frac{3}{40}) + \dots + f (\frac{39}{40}) - f (\frac{1}{2})$ is_________

A 19
B 12
C 10
D 8

Verified Solution

Answer & Solution

Correct Answer

(A) 19

Step-by-step Solution

Detailed explanation

f (x) = \frac{4^{x}}{4^{x} + 2}, f (1 - x) = \frac{4^{1 - x}}{4^{1 - x} + 2} = \frac{2}{4^{x} + 2}

\Rightarrow f (x) + f (1 - x) = 1

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From JEE Advanced

Let $z=x+i y$ be a complex number, where $x$ and $y$ are integers. Then, the area of the rectangle whose vertices are the roots of the equation $z \bar{z}^3+\bar{z} z^3=350$ is

JEE Advanced 2009 Medium

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists :

List - i	List - II
A. Alpha decay	P. ${ }_8^{15} O \rightarrow{ }_7^{15} N+\ldots \ldots \ldots$.
B. $\beta^{+}$decay	Q. ${ }_{92}^{238} U \rightarrow{ }_{90}^{234} Th +\ldots \ldots \ldots$.
C. Fission	R. ${ }_{83}^{185} Bi \rightarrow{ }_{82}^{184} Pb+\ldots \ldots \ldots$.
D. Proton emission	S. ${ }_{94}^{239} Pu \rightarrow{ }_{57}^{140} La +\ldots \ldots \ldots$.

JEE Advanced 2013 Medium

One mole of a monatomic ideal gas undergoes four thermodynamic processes as shown schematically in the PV - diagram below. Among these four processes, one is isobaric, one is isochoric, one is isothermal and one is adiabatic. Match the processes mentioned in List-1 with the corresponding statements in List-II.

	LIST -I		LIST -II
A)	In process $I$	P)	Work done by the gas is zero
B)	In process $I I$	Q)	Temperature of the gas remains unchanged
C)	In process $I I I$	R)	No heat is exchanged between the gas and its surroundings
D)	In process $I V$	S)	Work done by the gas is $6 P_{0} V_{0}$

JEE Advanced 2018 Medium

Let $w = \frac{\sqrt{3} + i}{2}$ and $P = \{w^{n} : n = 1, 2, 3, \dots\}$ .Further $H_{1} = \{z \in C : Re z > \frac{1}{2}\}$ and $H_{2} = \{z \in C : Re z < \frac{- 1}{2}\}$ , where C is the set of all complex numbers. If $z_{1} \in P \cap H_{1}, z_{2} \in P \cap H_{2}$ and O represents the origin, then $∠ z_{1} O z_{2} =$

JEE Advanced 2013 Hard

Explore more questions on app