JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f:[0, \infty) \rightarrow[0,3]\) be a function defined by \(f(x)=\max \{\sin t: 0 \leq t \leq x\}, \quad 0 \leq x \leq \pi\) \(\quad \quad \quad \quad \quad \quad 2+\cos x,\quad \quad \quad \quad x>\pi\) Then which of the following is true?
- A \(\mathrm{f}\) is differentiable everywhere in \((0, \infty)\)
- B \(\mathrm{f}\) is continuous everywhere but not differentiable exactly at two points in \((0, \infty)\)
- C \(\mathrm{f}\) is not continuous exactly at two points in \((0, \infty)\)
- D \(\mathrm{f}\) is continuous everywhere but not differentiable exactly at one point in \((0, \infty)\)
Answer & Solution
Correct Answer
(A) \(\mathrm{f}\) is differentiable everywhere in \((0, \infty)\)
Step-by-step Solution
Detailed explanation
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 the tangent at a point \(P\) on the parabola \(y ^2=3 x\) is parallel to the line \(x+2 y=1\) and the tangents at the points \(Q\) and \(R\) on the ellipse \(\frac{x^2}{4}+\frac{y^2}{1}=1\) are perpendicular to the line \(x-y=2\), then the area of the triangle \(PQR\) is:JEE Mains 2023 Hard
- Let \(P\) be an arbitrary point having sum of the squares of the distance from the planes \(x + y + z =0, l x - nz =0\) and \(x -2 y + z =0\) equal to \(9 .\) If the locus of the point \(P\) is \(x ^{2}+ y ^{2}+ z ^{2}=9,\) then the value of \(l- n\) is equal to ...... .JEE Mains 2021 Hard
- Let \(f(\mathrm{x})=\mathrm{x} \cos ^{-1}(-\sin |\mathrm{x}|), \quad \mathrm{x} \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right],\) then which of the following is true?JEE Mains 2020 Hard
- The area of the triangle with vertices \(A ( z ), B ( iz )\) and \(C(z+i z)\) isJEE Mains 2021 Medium
- Suppose \(A B\) is a focal chord of the parabola \(\mathrm{y}^2=12 \mathrm{x}\) of length \(l\) and slope \(\mathrm{m}<\sqrt{3}\). If the distance of the chord \(\mathrm{AB}\) from the origin is \(\mathrm{d}\), then \(l \mathrm{~d}^2\) is equal to ....................JEE Mains 2024 Medium
- A tangent to the hyperbola \(\frac{{{x^2}}}{4} - \frac{{{y^2}}}{2} = 1\) meets \(x-\) axis at \(P\) and \(y-\) axis at \(Q\). Lines \(PR\) and \(QR\) are drawn such that \(OPRQ\) is a rectangle (where \(O\) is the origin). Then \(R\) lies onJEE Mains 2013 Hard
More PYQs from JEE Mains
- Let the plane \(a x+b y+c z+d=0\) bisect the line joining the points \((4,-3,1)\) and \((2,3,-5)\) at the right angles. If \(a , b , c , d\) are integers, then the minimum value of \(\left(a^{2}+b^{2}+c^{2}+d^{2}\right)\) isJEE Mains 2021 Hard
- Let \(f (x)\) be a polynomial of degree \(4\) having extreme values at \(x\, = 1\) and \(x\, = 2\). If \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{f\left( x \right)}}{{{x^2}}} + 1} \right) = 3\) then \(f(-1)\) is equal toJEE Mains 2018 Hard
- Let \(f: R-\{0\} \rightarrow R\) be a function such that \(f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}\).
If the \(\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in R\), then \(\alpha+2 \beta\) is equal toJEE Mains 2025 Easy - If \(n\) is the number of irrational terms in the expansion of \(\left(3^{1 / 4}+5^{1 / 8}\right)^{60},\) then \(( n -1)\) is divisible byJEE Mains 2021 Hard
- Let the coefficients of \(x ^{-1}\) and \(x ^{-3}\) in the expansion of \(\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0\), be \(m\) and \(n\) respectively. If \(r\) is a positive integer such \(m n^{2}={ }^{15} C _{ r } .2^{ r }\), then the value of \(r\) is equal toJEE Mains 2022 Medium
- Let \(ABCD\) be a quadrilateral. If \(E\) and \(F\) are the mid points of the diagonals \(AC\) and \(BD\) respectively and \((\overrightarrow{ AB }-\overline{ BC })+(\overrightarrow{ AD }-\overrightarrow{ DC })= k \overline{ FE }\), then \(k\) is equal toJEE Mains 2023 Hard