if f(x)= \(\frac{(tan1\degree)x+\log_e(123)}{x\log_e(1234)-(tan1\degree)},x>0\) , then the least value of \(f(f(x))+f(f(\frac{4}{x}))\)is
- 0
- 2
- 4
- 8
The Correct Option is C
Solution and Explanation
Given:
\( f(x) = \frac{(\tan 1)x + \log_{12}3}{x \log_{12}34 - \tan 1} \)
Let:
\( A = \tan 1, \quad B = \log_{12}3, \quad C = \log_{12}34 \)
Rewriting \(f(x)\):
\( f(x) = \frac{Ax + B}{xC - A} \)
Step 1: Evaluate \(f(f(x))\):
\( f(f(x)) = f \left( \frac{Ax+B}{xC-A} \right) \)
Substitute \(f(x)\) into \(f\):
\( f(f(x)) = \frac{A \left( \frac{Ax+B}{xC-A} \right) + B}{C \left( \frac{Ax+B}{xC-A} \right) - A} \)
Simplify:
\( f(f(x)) = \frac{A(Ax+B) + B(xC - A)}{C(Ax+B) - A(xC - A)} \)
Expand numerator and denominator:
Numerator: \( A^2x + AB + xBC - AB = x(A^2 + BC) \)
Denominator: \( ACx + BC - ACx + A^2 = A^2 + BC \)
Thus:
\( f(f(x)) = \frac{x(A^2 + BC)}{A^2 + BC} = x \)
Step 2: Evaluate \(f(f(4/x))\):
\( f(f(4/x)) = \frac{4}{x} \)
Step 3: Evaluate \(f(f(x)) + f(f(4/x))\):
\( f(f(x)) + f(f(4/x)) = x + \frac{4}{x} \)
Step 4: Apply AM-GM Inequality: Using AM-GM:
\( \frac{x + \frac{4}{x}}{2} \geq \sqrt{x \cdot \frac{4}{x}} \)
\( x + \frac{4}{x} \geq 4 \)
Final Answer: \( x + \frac{4}{x} \geq 4 \)
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Trigonometric Equations Questions
Let \(S=\left\{0∈(0,\frac{π}{2}) : \sum^{9}_{m=1} \sec(θ+(m-1)\frac{π}{6})\sec(θ+\frac{mπ}{6}) = -\frac{8}{\sqrt3}\right\}\)
Then,- The number of solution of the equation \(4sin^2 x-4cos^3 x+9-4cos x = 0\), \(x ∈ [-2\pi, 2\pi]\)
- The number of solution of equation \(e^{sinx} - 2e ^{-sinx} = 2\) is
- If\(\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}\)is the solution of\( 4cos\theta+ 5sin\theta=1\)then the value of \(tan\alpha\) is
- If \(\tan A = \frac{1}{\sqrt{x(x^2 + x + 1)}}, \quad \tan B = \frac{\sqrt{x}}{\sqrt{x^2 + x + 1}}\) and \(\tan C = \left(x^3 + x^2 + x^{-1}\right)^{\frac{1}{2}}, \quad 0 < A, B, C < \frac{\pi}{2}\),then \( A + B \) is equal to:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.