Question:
The minimum value of the function $f(x) = x^2 + \frac{250}{x}$ for $x > 0$ is:
The minimum value of the function $f(x) = x^2 + \frac{250}{x}$ for $x > 0$ is:
Show Hint
Alternatively, you can use the AM-GM inequality: $x^2 + \frac{125}{x} + \frac{125}{x} \ge 3\sqrt[3]{x^2 \cdot \frac{125}{x} \cdot \frac{125}{x}} = 3(25) = 75$. This avoids calculus entirely!
Updated On: May 31, 2026
- $75$
- $50$
- $25$
- $100$
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Concept
To find the local extremum of a function $f(x)$, we find the critical points where $f'(x) = 0$ and use the second derivative test $f''(x) > 0$ to confirm a local minimum.
Step 2: Meaning
We find the first derivative $f'(x)$ of the function, set it to zero to solve for the critical value of $x$, and calculate $f(x)$ at that point.
Step 3: Analysis
The function is: \[ f(x) = x^2 + \frac{250}{x} \] Differentiating with respect to $x$: \[ f'(x) = 2x - \frac{250}{x^2} \] Setting $f'(x) = 0$: \[ 2x = \frac{250}{x^2} \implies 2x^3 = 250 \implies x^3 = 125 \implies x = 5 \] Finding the second derivative to verify the minimum: \[ f''(x) = 2 + \frac{500}{x^3} \] At $x = 5$, $f''(5) = 2 + \frac{500}{125} = 6 > 0$, confirming a minimum. Now, calculate the minimum value at $x = 5$: \[ f(5) = 5^2 + \frac{250}{5} = 25 + 50 = 75 \]
Step 4: Conclusion
The minimum value of the given function for $x > 0$ is $75$. Final Answer: (A)
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Maxima and Minima Questions
- If \( x \) is real and \( \alpha, \beta \) are maximum and minimum values of \( \frac{x^2 - x + 1}{x^2 + x + 1} \) respectively, then \( \alpha + \beta = \):
- The sum of the global minimum and global maximum values of the function \[ f(x) = \frac{4}{3}x^3 - 4x \quad \text{in } [0, 2] \text{ is} \]
- The maximum value of \( x^4 y^4 \) when \( a^2 x^4 + b^2 y^4 = c^6 \) is:
- If \( A = \left\{ \frac{x}{9x^2 + 20} \right\} \) and \( f: A \to \mathbb{R} \) is defined by \( f(x) = 2x^3 - 15x^2 + 36x - 48 \), then the maximum value of \( f(x) \) is
- If \( f(x) = 2x^3 - 3x^2 + 4 \), then the minimum value of \( f(x) \) in the interval \( [0, 3] \) is:
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions