Question:
If Rolle's theorem holds for the function \(f(x)=x^{2}-px+10\), \(x\in [0,4]\) with \(c=2\), then \(p=\)
If Rolle's theorem holds for the function \(f(x)=x^{2}-px+10\), \(x\in [0,4]\) with \(c=2\), then \(p=\)
Show Hint
Rolle's theorem needs f(0) = f(4) and f'(c) = 0; either condition gives p.
Updated On: Nov 12, 2026
- \(4\)
- \(2\)
- \(-4\)
- \(-2\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
Rolle's theorem requires \(f(0)=f(4)\) and some \(c\) with \(f'(c)=0\).
Step 2: Use f(0) = f(4)
\(f(0)=10\) and \(f(4)=16-4p+10=26-4p\).
Setting \(26-4p=10\) gives \(p=4\).
Step 3: Verify c = 2
\(f'(x)=2x-p\), so \(f'(2)=4-p=0\) gives \(p=4\). Both conditions agree.
Final Answer:
The value of \(p\) is 4. \[ \boxed{\text{(A) }4} \]
Rolle's theorem requires \(f(0)=f(4)\) and some \(c\) with \(f'(c)=0\).
Step 2: Use f(0) = f(4)
\(f(0)=10\) and \(f(4)=16-4p+10=26-4p\).
Setting \(26-4p=10\) gives \(p=4\).
Step 3: Verify c = 2
\(f'(x)=2x-p\), so \(f'(2)=4-p=0\) gives \(p=4\). Both conditions agree.
Final Answer:
The value of \(p\) is 4. \[ \boxed{\text{(A) }4} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Mean Value Theorem Questions
- The value of c of Lagrange's mean value theorem for $f(x)=\sqrt{25-x^{2}}$ on $[1,5]$ is
- The abscissae of the points of the curve \[ y = x^3 \quad \text{are in the interval} \quad [-2, 2], \] where the slope of the tangents can be obtained by the mean value theorem for the interval \( [-2, 2] \), are
- If the function \(f(x) = ax^3+bx^2+11x-6\), defined on \([1,3]\), satisfies all the conditions of Rolle's theorem for \(c = 2+\frac{1}{\sqrt{3}}\), then
- If Rolle's theorem is applicable for the function \(f(x) = log(\frac{x^2+a}{x})\) on \([3,4]\) with \(c\in (3,4)\) such that \(f^'(c) = 0\), then the value of \(f^{''}(c)\) is
- Let \(f(x)\) be the differentiable function for all \(x\) such that \(f^'(x)\leq 5\) and \(f(1) = 4\). The maximum value of \(f(5)\) is...
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions