Question:
If m is the minimum value of k for which the function \(f(x) = x\sqrt{kx - x^2}\) is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :
If m is the minimum value of k for which the function \(f(x) = x\sqrt{kx - x^2}\) is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :
Show Hint
Compute \(f'(x)=\frac{x(3k-4x)}{2\sqrt{kx-x^2}}\) and require it to be non-negative on (0,3), so \(3k\ge 4x\) for all x up to 3.
Updated On: Nov 5, 2026
- \(\left(5, 3\sqrt{6}\right)\)
- \(\left(4, 3\sqrt{3}\right)\)
- \(\left(4, 3\sqrt{2}\right)\)
- \(\left(3, 3\sqrt{3}\right)\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
\(f'(x)=\sqrt{kx-x^2}+\frac{x(k-2x)}{2\sqrt{kx-x^2}}=\frac{2(kx-x^2)+kx-2x^2}{2\sqrt{kx-x^2}}=\frac{x(3k-4x)}{2\sqrt{kx-x^2}}\).
For \(x\in(0,3)\), \(f'\ge0\) needs \(3k\ge4x\) for all such x, i.e. \(3k\ge12\Rightarrow k\ge4\). (The domain condition \(kx-x^2\ge0\) on [0,3] only needs \(k\ge3\), which is weaker.) So \(m=4\).
With \(k=4\), f is increasing on [0,3], so its maximum is at \(x=3\): \(M=3\sqrt{12-9}=3\sqrt3\).
Ordered pair \((4,3\sqrt3)\).
Answer: (2) \((4, 3\sqrt{3})\)
For \(x\in(0,3)\), \(f'\ge0\) needs \(3k\ge4x\) for all such x, i.e. \(3k\ge12\Rightarrow k\ge4\). (The domain condition \(kx-x^2\ge0\) on [0,3] only needs \(k\ge3\), which is weaker.) So \(m=4\).
With \(k=4\), f is increasing on [0,3], so its maximum is at \(x=3\): \(M=3\sqrt{12-9}=3\sqrt3\).
Ordered pair \((4,3\sqrt3)\).
Answer: (2) \((4, 3\sqrt{3})\)
Was this answer helpful?
0
0
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_____
View More Questions
Top JEE Main Applications of Derivatives Questions
- Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) + 3f\left( \frac{\pi}{2} - x \right) = \sin x, x \in \mathbb{R}$. Let the maximum value of $f$ on $\mathbb{R}$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3, \beta>0$, is $\alpha^2$, then $30\beta^3$ is equal to :}
- Statement-1: The function \(x^2(e^x+e^{-x})\) is increasing for all \(x \gt 0\).
Statement-2: The functions \(x^2e^x\) and \(x^2e^{-x}\) are increasing for all \(x \gt 0\) and the sum of two increasing functions in any interval \((a,b)\) is an increasing function in \((a,b)\). - For the function
\(f(x) = 4\log_e(x-1) - 2x^2 + 4x + 5,\ x \gt 1\)
, which one of the following is NOT correct? - Let \(f\) be a real valued function, defined on \(\mathbb{R} - \{-1, 1\}\) and given by
\(f(x) = 3\log_e\left|\frac{x-1}{x+1}\right| - \frac{2}{x-1}\).
Then in which of the following intervals, function \(f(x)\) is increasing? - Let \(a\) be an integer such that all the real roots of the polynomial \(2x^5+5x^4+10x^3+10x^2+10x+10\) lie in the interval \((a,a+1)\). Then, \(|a|\) is equal to ___________.
View More Questions
Top JEE Main Questions
- 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 ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
View More Questions