JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f, g: R \rightarrow R\) be two real valued functions defined as \(f(x)=\left\{\begin{array}{cl}-|x+3| & , \quad x<0 \\ e^{x} & , \quad x \geq 0\end{array}\right.\) and \(g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x & , \quad x<0 \\ 4 x+k_{2} & , \quad x \geq 0\end{array}\right.\), where \(k_{1}\) and \(k_{2}\) are real constants. If \((gof)\) is differentiable at \(x=0\), then \((gof) (-4)+(gof)\, (4)\) is equal to
- A \(4\left(e^{4}+1\right)\)
- B \(2\left(2 e ^{4}+1\right)\)
- C \(4 e ^{4}\)
- D \(2\left(2 e ^{4}-1\right)\)
Answer & Solution
Correct Answer
(D) \(2\left(2 e ^{4}-1\right)\)
Step-by-step Solution
Detailed explanation
\(f(x)=\left\{\begin{array}{lll}x+3 & ; & x<-3 \\ -(x+3) & ; & -3 \leq x<0 \\ e^{x} & ; & x \geq 0\end{array}\right\}\) \(g(x)=\left\{\begin{array}{lll}x^{2}+k_{1} x & ; & x<0 \\ 4 x+k_{2} & ; & x \geq 0\end{array}\right\}\)…
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 \(\lim _{x \rightarrow 0} \frac{\cos (2 x)+a \cos (4 x)-b}{x^4}\) is finite, then \((a+b)\) is equal to :JEE Mains 2025 Medium
- Let \(S = \{x \in [-\pi, \pi] : \sin x (\sin x + \cos x) = a, a \in \mathbb{Z}\}\). Then \(n(S)\) is equal to :JEE Mains 2026 Medium
- Let \(f(x)\) and \(g(x)\) be twice differentiable functions satisfying \(f''(x) = g''(x)\) for all \(x \in \mathbf{R}\), \(f'(1) = 2g'(1) = 4\) and \(g(2) = 3f(2) = 9\). Then \(f(25) - g(25)\) is equal to :JEE Mains 2026 Medium
- If \(\sum_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in \mathbf{Z}\), then \(a^2+b^2\) is equal to :JEE Mains 2025 Hard
- Let \({a_1},{a_2},\;.\;.\;.\;.,{a_{49}}\) be in \(A.P.\) such that \(\mathop \sum \limits_{k = 0}^{12} {a_{4k + 1}} = 416\) and \({a_9} + {a_{43}} = 66\). If \(a_1^2 + a_2^2 + \ldots + a_{17}^2 = 140m,\) then \(m = \;\;..\;.\;.\;.\;\)JEE Mains 2018 Hard
- If \(\alpha = {\cos ^{ - 1}}\,\left( {\frac{3}{5}} \right),\beta = {\tan ^{ - 1}}\,\left( {\frac{1}{3}} \right)\), where \(0 < \alpha ,\beta < \frac{\pi }{2}\), then \(\alpha - \beta \) is equal toJEE Mains 2019 Hard
More PYQs from JEE Mains
- \(\lim _{x \rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6} x^3\)JEE Mains 2023 Hard
- If the system of equations \(\mathrm{x}+4 \mathrm{y}-\mathrm{z}=\lambda\), \(7 x+9 y+\mu z=-3,5 x+y+2 z=-1\) has infinitely many solutions, then \((2 \mu .+3 \lambda)\) is equal to :JEE Mains 2024 Medium
- \(\smallint \frac{{dx}}{{{x^2}{{\left( {{x^4} + 1} \right)}^{\frac{3}{4}}}}} = \)JEE Mains 2015 Hard
- Let \(O\) be the origin, and \(\mathrm{M}\) and \(\mathrm{N}\) be the points on the lines \(\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}\) and \(\frac{\mathrm{x}+8}{12}=\frac{\mathrm{y}+2}{5}=\frac{\mathrm{z}+11}{9}\) respectively such that \(\mathrm{MN}\) is the shortest distance between the given lines. Then \(\overrightarrow{\mathrm{OM}} \cdot \overrightarrow{\mathrm{ON}}\) is equal to ...........JEE Mains 2024 Hard
- Let the function \(f(x)=\left(x^2+1\right)\left|x^2-a x+2\right|+\cos |x|\) be not differentiable at the two points \(x=\alpha=2\) and \(x=\beta\). Then the distance of the point \((\alpha, \beta)\) from the line \(12 x+5 y+10=0\) is equal to :JEE Mains 2025 Hard
- Let \(y=y(x)\) be the solution of the differential equation \(\sec ^2 x d x+\left(e^{2 y} \tan ^2 x+\tan x\right) d y=0 \) , \(0 < x < \frac{\pi}{2}, y\left(\frac{\pi}{4}\right)=0\). If \(y\left(\frac{\pi}{6}\right)=\alpha\), Then \(\mathrm{e}^{8 \alpha}\) is equal to ...........JEE Mains 2024 Hard