JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(\mathrm{f}\) be any continuous function on \([0,2]\) and twice differentiable on \((0,2)\). If \(\mathrm{f}(0)=0, \mathrm{f}(1)=1\) and \(f(2)=2\), then
- A \(f^{\prime \prime}(x)=0\) for all \(x \in(0,2)\)
- B \(f^{\prime \prime}(x)=0\) for some \(x \in(0,2)\)
- C \(f^{\prime}(x)=0\) for some \(x \in[0,2]\)
- D \(f^{\prime \prime}(x)>0\) for all \(x \in(0,2)\)
Answer & Solution
Correct Answer
(B) \(f^{\prime \prime}(x)=0\) for some \(x \in(0,2)\)
Step-by-step Solution
Detailed explanation
\(f(0)=0 \quad f(1)=1\) and \(f(2)=2\) Let \(\mathrm{h}(\mathrm{x})=f(\mathrm{x})-\mathrm{x}\) has three roots By Rolle's theorem \(\mathrm{h}^{\prime}(\mathrm{x})=f^{\prime}(\mathrm{x})-1\) has at least two roots…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(f(x)\, = {x^2} - x + 5,\,\,x > \frac{1}{2},\) and \(g(x)\) is its inverse function, then \(g'(7)\) equalsJEE Mains 2014 Hard
- Two poles standing on a horizontal ground are of heights \(5\,m\) and \(10\, m\) respectively. The line joining their tops makes an angle of \(15^o\) with ground. Then the distance (in \(m\)) between the poles, isJEE Mains 2019 Hard
- Let \(S\) be the set of all \((\alpha, \beta), \pi<\alpha, \beta<2 \pi\), for which the complex number \(\frac{1-i \sin \alpha}{1+2 i \sin \alpha}\) is purely imaginary and \(\frac{1+i \cos \beta}{1-2 i \cos \beta}\) is purely real. Let \(Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S\) . Then \(\sum_{(\alpha, \beta) \in s }\left(i Z_{\alpha \beta}+\frac{1}{i \bar{Z}_{\alpha \beta}}\right)\) is equal to.JEE Mains 2022 Hard
- Let for two distinct values of \(p\) the lines \(y=x+p\) touch the ellipse \(\mathrm{E}: \frac{\mathrm{x}^2}{4^2}+\frac{\mathrm{y}^2}{3^2}=1\) at the points A and B . Let the line \(\mathrm{y}=\mathrm{x}\) intersect E at the points C and \(D\). Then the area of the quadrilateral \(A B C D\) is equal toJEE Mains 2025 Medium
- If the foot of the perpendicular drawn from \((1,9\), 7) to the line passing through the point \((3,2,1)\) and parallel to the planes \(x+2 y+z=0\) and \(3 y-z=3\) is \((\alpha, \beta, \gamma)\), then \(\alpha+\beta+\gamma\) is equal toJEE Mains 2023 Hard
- The sum of the series
\(2 \times 1 \times{ }^{20} \mathrm{C}_4-3 \times 2 \times{ }^{20} \mathrm{C}_5+4 \times 3 \times{ }^{20} \mathrm{C}_6-5 \times 4\) \(\times { }^{20} \mathrm{C}_7+\ldots+18 \times 17 \times{ }^{20} \mathrm{C}_{20}\), is equal toJEE Mains 2025 Medium
More PYQs from JEE Mains
- Let \(P(\alpha,\beta,\gamma)\) be the point on the line \(\frac{x-1}{2}=\frac{y+1}{-3}=z\) at a distance \(4\sqrt{14}\) from the point (1, -1, 0) and nearer to the origin. Then the shortest distance, between the lines \(\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}\) and \(\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}\), is equal toJEE Mains 2026 Easy
- Let the line \(\ell: x =\frac{1- y }{-2}=\frac{ z -3}{\lambda}, \lambda \in R\) meet the plane \(P : x +2 y +3 z =4\) at the point \((\alpha, \beta, \gamma)\). If the angle between the line \(\ell\) and the plane \(P\) is \(\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)\), then \(\alpha+2 \beta+6 \gamma\) is equal toJEE Mains 2023 Hard
- A random variable \(X\) has the following probability distribution
Then \(\mathrm{P}(\mathrm{X}> 2)\) is equal to\(X\) \(1\) \(2\) \(3\) \(4\) \(5\) \(P(X)\) \(K^2\) \(2K\) \(K\) \(2K\) \(5K^2\) JEE Mains 2020 Hard - Shortest distance between the lines \(\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}\) and \(\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}\) isJEE Mains 2023 Easy
- If the point \((1, 4)\) lies inside the circle \(x^2 + y^2-6x - 10y + p = 0\) and the circle does not touch or intersect the coordinate axes, then the set of all possible values of \(p\) is the intervalJEE Mains 2014 Hard
- Let the locus of the centre \((\alpha, \beta), \beta>0\), of the circle which touches the circle \(x ^{2}+( y -1)^{2}=1\) externally and also touches the \(x\)-axis be \(L\). Then the area bounded by \(L\) and the line \(y =4\) is.JEE Mains 2022 Hard