Top 50 Most Repeated CONTINUITY AND DIFFERENTIABILITY PYQs | JEE MAINS
A curated collection of the most important questions from CONTINUITY AND DIFFERENTIABILITY, fully solved with step-by-step concepts to prepare for JEE MAINS.
A curated collection of the most important questions from CONTINUITY AND DIFFERENTIABILITY, fully solved with step-by-step concepts to prepare for JEE MAINS.
Chain rule: derivative of outer log, then sin, then exp....
Read Full Step-by-Step Solution →Sum of |x−a| functions is non-differentiable at points where argument is zero....
Read Full Step-by-Step Solution →Differentiate term‑by‑term: $\frac{d}{dx}e^{\sin x}=e^{\sin x}\cos x$ and $\frac{d}{dx}\ln(1+\cos x)=\frac{-\sin x}{1+\cos x}$. At $x=0$, $e^{\sin 0}=...
Read Full Step-by-Step Solution →Limit exists and derivative at 0 is 0 by definition....
Read Full Step-by-Step Solution →First compute $\frac{dy}{d\theta} = 9\sin^2\theta\cos\theta$, $\frac{dx}{d\theta} = -6\cos^2\theta\sin\theta$. Then $\frac{dy}{dx} = \frac{dy/d\theta}...
Read Full Step-by-Step Solution →Continuity at \(x=2\) requires the left‑hand limit to equal the right‑hand value. Setting \(4a+6 = 5(2)-7 = 3\) gives \(a = -3/4 = -0.75\)....
Read Full Step-by-Step Solution →Differentiability requires continuity and equality of the left‑ and right‑hand derivatives at \(x=2\). Continuity gives \(4 = 2a + b\). The left deriv...
Read Full Step-by-Step Solution →By squeeze theorem, $|x^2\sin(1/x)| \leq x^2 \to 0$, so limit is 0, matching $f(0)$....
Read Full Step-by-Step Solution →Lagrange's MVT requires $f'(c) = \frac{f(3)-f(1)}{3-1}$. Solving gives $c=2$....
Read Full Step-by-Step Solution →First derivative: $\frac{dy}{dx} = \cos(\sin x) \cdot \cos x$. Second derivative: Use product rule: $\frac{d^2y}{dx^2} = -\sin(\sin x)\cdot \cos^2 x -...
Read Full Step-by-Step Solution →Simplify $f(x) = x+2$ for $x \ne 2$, so limit is 4....
Read Full Step-by-Step Solution →To ensure differentiability at x = 1, both continuity and derivative conditions must be satisfied. Continuity gives b + c = 1, while differentiability...
Read Full Step-by-Step Solution →To ensure differentiability, first check continuity by equating limits at x = 1 (6 = a + b). Then equate left and right derivatives (5 = a). Solving t...
Read Full Step-by-Step Solution →The function f(x) = |x| is continuous at x = 0 because lim(x→0) f(x) = f(0) = 0, but it is not differentiable at x = 0 because the left and right limi...
Read Full Step-by-Step Solution →Continuity requires 3 + a = 3 + b ⇒ a = b. Differentiability requires 4 = b. Thus, a = 4....
Read Full Step-by-Step Solution →Chain rule: derivative of outer log, then sin, then exp....
Read Full Step-by-Step Solution →Sum of |x−a| functions is non-differentiable at points where argument is zero....
Read Full Step-by-Step Solution →Differentiate term‑by‑term: $\frac{d}{dx}e^{\sin x}=e^{\sin x}\cos x$ and $\frac{d}{dx}\ln(1+\cos x)=\frac{-\sin x}{1+\cos x}$. At $x=0$, $e^{\sin 0}=...
Read Full Step-by-Step Solution →Limit exists and derivative at 0 is 0 by definition....
Read Full Step-by-Step Solution →First compute $\frac{dy}{d\theta} = 9\sin^2\theta\cos\theta$, $\frac{dx}{d\theta} = -6\cos^2\theta\sin\theta$. Then $\frac{dy}{dx} = \frac{dy/d\theta}...
Read Full Step-by-Step Solution →Continuity at \(x=2\) requires the left‑hand limit to equal the right‑hand value. Setting \(4a+6 = 5(2)-7 = 3\) gives \(a = -3/4 = -0.75\)....
Read Full Step-by-Step Solution →Differentiability requires continuity and equality of the left‑ and right‑hand derivatives at \(x=2\). Continuity gives \(4 = 2a + b\). The left deriv...
Read Full Step-by-Step Solution →By squeeze theorem, $|x^2\sin(1/x)| \leq x^2 \to 0$, so limit is 0, matching $f(0)$....
Read Full Step-by-Step Solution →Lagrange's MVT requires $f'(c) = \frac{f(3)-f(1)}{3-1}$. Solving gives $c=2$....
Read Full Step-by-Step Solution →First derivative: $\frac{dy}{dx} = \cos(\sin x) \cdot \cos x$. Second derivative: Use product rule: $\frac{d^2y}{dx^2} = -\sin(\sin x)\cdot \cos^2 x -...
Read Full Step-by-Step Solution →Simplify $f(x) = x+2$ for $x \ne 2$, so limit is 4....
Read Full Step-by-Step Solution →To ensure differentiability at x = 1, both continuity and derivative conditions must be satisfied. Continuity gives b + c = 1, while differentiability...
Read Full Step-by-Step Solution →To ensure differentiability, first check continuity by equating limits at x = 1 (6 = a + b). Then equate left and right derivatives (5 = a). Solving t...
Read Full Step-by-Step Solution →The function f(x) = |x| is continuous at x = 0 because lim(x→0) f(x) = f(0) = 0, but it is not differentiable at x = 0 because the left and right limi...
Read Full Step-by-Step Solution →Continuity requires 3 + a = 3 + b ⇒ a = b. Differentiability requires 4 = b. Thus, a = 4....
Read Full Step-by-Step Solution →Chain rule: derivative of outer log, then sin, then exp....
Read Full Step-by-Step Solution →Sum of |x−a| functions is non-differentiable at points where argument is zero....
Read Full Step-by-Step Solution →Differentiate term‑by‑term: $\frac{d}{dx}e^{\sin x}=e^{\sin x}\cos x$ and $\frac{d}{dx}\ln(1+\cos x)=\frac{-\sin x}{1+\cos x}$. At $x=0$, $e^{\sin 0}=...
Read Full Step-by-Step Solution →Limit exists and derivative at 0 is 0 by definition....
Read Full Step-by-Step Solution →First compute $\frac{dy}{d\theta} = 9\sin^2\theta\cos\theta$, $\frac{dx}{d\theta} = -6\cos^2\theta\sin\theta$. Then $\frac{dy}{dx} = \frac{dy/d\theta}...
Read Full Step-by-Step Solution →Continuity at \(x=2\) requires the left‑hand limit to equal the right‑hand value. Setting \(4a+6 = 5(2)-7 = 3\) gives \(a = -3/4 = -0.75\)....
Read Full Step-by-Step Solution →Differentiability requires continuity and equality of the left‑ and right‑hand derivatives at \(x=2\). Continuity gives \(4 = 2a + b\). The left deriv...
Read Full Step-by-Step Solution →By squeeze theorem, $|x^2\sin(1/x)| \leq x^2 \to 0$, so limit is 0, matching $f(0)$....
Read Full Step-by-Step Solution →Lagrange's MVT requires $f'(c) = \frac{f(3)-f(1)}{3-1}$. Solving gives $c=2$....
Read Full Step-by-Step Solution →First derivative: $\frac{dy}{dx} = \cos(\sin x) \cdot \cos x$. Second derivative: Use product rule: $\frac{d^2y}{dx^2} = -\sin(\sin x)\cdot \cos^2 x -...
Read Full Step-by-Step Solution →Simplify $f(x) = x+2$ for $x \ne 2$, so limit is 4....
Read Full Step-by-Step Solution →To ensure differentiability at x = 1, both continuity and derivative conditions must be satisfied. Continuity gives b + c = 1, while differentiability...
Read Full Step-by-Step Solution →To ensure differentiability, first check continuity by equating limits at x = 1 (6 = a + b). Then equate left and right derivatives (5 = a). Solving t...
Read Full Step-by-Step Solution →The function f(x) = |x| is continuous at x = 0 because lim(x→0) f(x) = f(0) = 0, but it is not differentiable at x = 0 because the left and right limi...
Read Full Step-by-Step Solution →Continuity requires 3 + a = 3 + b ⇒ a = b. Differentiability requires 4 = b. Thus, a = 4....
Read Full Step-by-Step Solution →Chain rule: derivative of outer log, then sin, then exp....
Read Full Step-by-Step Solution →