JEE Mains · Maths · STD 12 - 5. continuity and differentiation
If \(f(x)=\int_{0}^{x}(5+|1-t|) d t, \quad x>2\) \(\quad \quad \quad \quad \quad 5 x+1,\quad \quad \quad \quad \quad x \leq 2\), then
- A \(f(x)\) is not differentiable at \(x=1\)
- B \(f(x)\) is continuous but not differentiable at \(x=2\)
- C \(f(x)\) is not continuous at \(x=2\)
- D \(f(x)\) is everywhere differentiable
Answer & Solution
Correct Answer
(B) \(f(x)\) is continuous but not differentiable at \(x=2\)
Step-by-step Solution
Detailed explanation
\(f(x)=\int_{0}^{1}(5+(1-t)) d t+\int_{1}^{x}(5+(t-1)) d t\) \(=6-\frac{1}{2}+\left.\left(4 t+\frac{t^{2}}{2}\right)\right|_{1} ^{x}\) \(=\frac{11}{2}+4 x+\frac{x^{2}}{2}-4-\frac{1}{2}\) \(=\frac{x^{2}}{2}+4 x+1\) \(f\left(2^{+}\right)=2+8+1=11\) \(\Rightarrow\) continuous at…
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
- Let \(H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1\) be the hyperbola, whose eccentricity is \(\sqrt{3}\) and the length of the latus rectum is \(4 \sqrt{3}\). Suppose the point \((\alpha, 6), \alpha>0\) lies on \(H\). If \(\beta\) is the product of the focal distances of the point \((\alpha, 6)\), then \(\alpha^2+\beta\) is equal to :JEE Mains 2024 Hard
- Two cards are drawn successively with replacement from a well shuffled deck of \(52\) cards. Let \(X\) denote the random variable of number of aces obtained in the two drawn cards. Then \(P\,\left( {X = 1} \right)\, + P\,\left( {X = 2} \right)\) equalsJEE Mains 2019 Hard
- A coin is tossed \(8\) times. If the probability that exactly \(4\) heads appear in the first six tosses and exactly \(3\) heads appear in the last five tosses is \(p\), then \(96p\) is equal to _____.JEE Mains 2026 Hard
- If \([.]\) represents the greatest integer function, then the value of \(\int_{0}^{\sqrt{\pi / 2}}\left(\left[ x ^{2}\right]+[-\cos x ]\right) d x\) is.............JEE Mains 2021 Hard
- Let A be the point \((3, 0)\) and circles with variable diameter AB touch the circle \(x^2 + y^2 = 36\) internally. Let the curve C be the locus of the point B. If the eccentricity of C is \(e\), then \(72e^2\) is equal to _______.JEE Mains 2026 Hard
- Let \(f(x)\) be a quadratic polynomial such that \(f(-2)\) \(+f(3)=0\). If one of the roots of \(f(x)=0\) is \(-1\), then the sum of the roots of \(f(x)=0\) is equal toJEE Mains 2022 Hard
More PYQs from JEE Mains
- Let \(\alpha\) be the angle between the lines whose direction cosines satisfy the equations \(l+m-n=0\) and \(l^{2}+m^{2}-n^{2}=0 .\) Then the value of \(\sin ^{4} \alpha+\cos ^{4} \alpha\) isJEE Mains 2021 Hard
- The number of three-digit even numbers, formed by the digits \(0,1,3,4,6,7\) if the repetition of digits is not allowed, is .... .JEE Mains 2021 Medium
- For \(p\,>\,0\), a vector \(\vec{v}_{2}=2 \hat{i}+(p+1) \hat{j}\) is obtained by rotating the vector \(\vec{v}_{1}=\sqrt{3} p \hat{i}+\hat{j}\) by an angle \(\theta\) about origin in counter clockwise direction. If \(\tan \theta=\frac{(\alpha \sqrt{3}-2)}{4 \sqrt{3}+3}\), then the value of \(\alpha\) is equal to \(....\)JEE Mains 2021 Hard
- Let \(a , b , c\) and \(d\) be positive real numbers such that \(a+b+c+d=11\). If the maximum value of \(a^5 b^3 c^2 d\) is \(3750 \beta\), then the value of \(\beta\) isJEE Mains 2023 Hard
- Let \(a , b , c\) be three distinct positive real numbers such that \((2 a)^{\log _{\varepsilon} a}=(b c)^{\log _e b}\) and \(b^{\log _e 2}=a^{\log _e c}\). Then \(6 a+5 b c\) is equal to \(........\).JEE Mains 2023 Hard
- A symmetrical form of the line of intersection of the planes \(x = ay + b\) and \(z = cy + d\) isJEE Mains 2014 Medium