Question:
The maximum value of the function $f(x)=x\sqrt{4x-x^{2}}$ is ________.
The maximum value of the function $f(x)=x\sqrt{4x-x^{2}}$ is ________.
Show Hint
Check values at critical points and boundaries for absolute maxima.
Updated On: Jun 26, 2026
- $\sqrt{3}$
- $4\sqrt{3}$
- $5\sqrt{3}$
- $3\sqrt{3}$
- $6\sqrt{3}$
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Concept
Find the derivative $f'(x)$ and equate it to zero to find critical points.
Step 2: Meaning
$f(x) = x(4x - x^2)^{1/2}$. Domain is $[0, 4]$.
Step 3: Analysis
$f'(x) = \sqrt{4x-x^2} + x \cdot \frac{1}{2\sqrt{4x-x^2}}(4-2x) = 0$.
$(4x-x^2) + x(2-x) = 0 \implies 4x - x^2 + 2x - x^2 = 0 \implies 6x - 2x^2 = 0 \implies x = 3$.
Step 4: Conclusion
At $x=3$, $f(3) = 3\sqrt{4(3)-(3)^2} = 3\sqrt{12-9} = 3\sqrt{3}$. Final Answer: (D)
Find the derivative $f'(x)$ and equate it to zero to find critical points.
Step 2: Meaning
$f(x) = x(4x - x^2)^{1/2}$. Domain is $[0, 4]$.
Step 3: Analysis
$f'(x) = \sqrt{4x-x^2} + x \cdot \frac{1}{2\sqrt{4x-x^2}}(4-2x) = 0$.
$(4x-x^2) + x(2-x) = 0 \implies 4x - x^2 + 2x - x^2 = 0 \implies 6x - 2x^2 = 0 \implies x = 3$.
Step 4: Conclusion
At $x=3$, $f(3) = 3\sqrt{4(3)-(3)^2} = 3\sqrt{12-9} = 3\sqrt{3}$. Final Answer: (D)
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Maxima and Minima Questions
- Let f(x)=πcosx+x2.The value of c∈(0,π) where f attains its local maximum/minumum is
- Let \( f(x) = x^2 + ax + \beta \). If \( f \) has a local minimum at \( (2,6) \), then \( f(0) \) is equal to
- The minimum of $f(x) = \dfrac{x^{100} - 1}{x^{100} + 1}, \; x \in \mathbb{R}$ is:
- Let \(f(x) = e^{x(2x^{2} + ax + 2 - a)}\). If \(f\) has a local minimum at \(x = 2\), the value of \(a\) is equal to
- If $f(x)=2x^{3}-3x^{2}-\lambda x+1, x\in[0,3]$ attains a local minimum at $x=2$, then the value of $\lambda$ is equal to
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions