JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Suppose for a differentiable function \(h, h(0)=0\), \(\mathrm{h}(1)=1\) and \(\mathrm{h}^{\prime}(0)=\mathrm{h}^{\prime}(1)=2\). If \(\mathrm{g}(\mathrm{x})=\mathrm{h}\left(\mathrm{e}^{\mathrm{x}}\right) \mathrm{e}^{\mathrm{h}(\mathrm{x})}\), then \(g^{\prime}(0)\) is equal to :
- A \(5\)
- B \(3\)
- C \(8\)
- D \(4\)
Answer & Solution
Correct Answer
(D) \(4\)
Step-by-step Solution
Detailed explanation
\( g(x)=h\left(e^x\right) \cdot e^{h(x)} \) \( g^{\prime}(x)=h\left(e^x\right) \cdot e^{h(x)} \cdot h^{\prime}(x)+e^{h(x)} h^{\prime}\left(e^x\right) \cdot e^x \) \( g^{\prime}(0)=h(1) e^{h(0)} h^{\prime}(0)+e^{h(0)} h^{\prime}(1) \) \( =2+2=4\)
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 \(f\) be any function continuous on \([\mathrm{a}, \mathrm{b}]\) and twice differentiable on \((a, b) .\) If for all \(x \in(a, b)\) \(f^{\prime}(\mathrm{x})>0\) and \(f^{\prime \prime}(\mathrm{x})<0,\) then for any \(\mathrm{c} \in(\mathrm{a}, \mathrm{b})\) \(\frac{f(\mathrm{c})-f(\mathrm{a})}{f(\mathrm{b})-f(\mathrm{c})}\) is greater thanJEE Mains 2020 Hard
- Let \(\alpha \) and \(\beta \) be the roots of equation \(p{x^2} + qx + r = 0\) ( where \(p \ne 0\)) . If \(p,q,r\) are in \(A.P.\) and \(\frac{1}{\alpha } + \frac{1}{\beta } = 4\) , then the value of \(\left| {\alpha - \beta } \right| \) isJEE Mains 2014 Hard
- Let \(\left(1+x+x^2\right)^{10}=a_0+a_1 x+a_2 x^2+\ldots .+a_{20} x^{20}\). If \(\left(a_1+a_3+a_5+\ldots .+a_{19}\right)-11 \mathrm{a}_2=121 \mathrm{k}\), then k is equal to ________.JEE Mains 2025 Medium
- Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}+\frac{\sqrt{2} y}{2 \cos ^{4} x-\cos 2 x}= Xe ^{\tan ^{-1}(\sqrt{2} \cot 2 x )}, 0 < x < \pi / 2\) with \(y\left(\frac{\pi}{4}\right)=\frac{\pi^{2}}{32}\). If \(y\left(\frac{\pi}{3}\right)=\frac{\pi^{2}}{18} e^{-\tan ^{-1}(\alpha)}\), then the value of \(3 \alpha^{2}\) is equal toJEE Mains 2022 Hard
- In a triangle \(A B C\), if \(\cos A+2 \cos B+\cos C=2\) and the lengths of the sides opposite to the angles \(A\) and \(C\) are \(3\) and \(7\) respectively, then \(\cos A-\cos\) \(C\) is equal toJEE Mains 2023 Hard
- Consider the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) having one of its focus at \(\mathrm{P}(-3,0)\). If the latus ractum through its other focus subtends a right angle at P and \(a^2 b^2=\alpha \sqrt{2}-\beta, \alpha, \beta \in \mathbb{N}\).JEE Mains 2025 Medium
More PYQs from JEE Mains
- Let a tangent to the Curve \(9 x^2+16 y^2=144\) intersect the coordinate axes at the points \(A\) and \(B\). Then, the minimum length of the line segment \(A B\) is \(.........\)JEE Mains 2023 Hard
- The distance of the point \((1,-5, 9)\) from the plane \(x - y + z = 5\) measured along the line \(x = y = z\) isJEE Mains 2016 Hard
- Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix, where \(a_{i j}= 1 , \quad\quad\text { if } i=j\) \(\quad\quad-x ,\quad \text { if }|i-j|=1\) \(\quad\quad2 x+1, \text { otherwise }\) Let a function f: \(\mathrm{R} \rightarrow \mathrm{R}\) be defined as \(\mathrm{f}(\mathrm{x})=\operatorname{det}(\mathrm{A})\). Then the sum of maximum and minimum values of \(f\) on \(R\) is equal to:JEE Mains 2021 Medium
- If the equation of the plane passing through the line of intersection of the planes \(2 x-y+z=3,4 x-3 y\) \(+5 z+9=0\) and parallel to the line \(\frac{x+1}{-2}=\frac{y+3}{4}=\frac{z-2}{5}\) is \(a x+b y+c z+6=0\). then a \(+ b + c\) is equal to \(.............\).JEE Mains 2023 Hard
- Let A be the set of all functions \(f: \mathbf{Z} \rightarrow \mathbf{Z}\) and R be a relation on A such that \(\mathrm{R}=\{(\mathrm{f}, \mathrm{g}): f(0)=\mathrm{g}(1)\) and \(f(1)=\mathrm{g}(0)\}\). Then R is:JEE Mains 2025 Medium
- Let \(z\) be those complex numbers which satisfy \(|z+5| \leq 4\) and \(z(1+i)+\bar{z}(1-i) \geq-10, i=\sqrt{-1}\) If the maximum value of \(Iz +\left.1\right|^{2}\) is \(\alpha+\beta \sqrt{2}\), then the value of \((\alpha+\beta)\) is ...... .JEE Mains 2021 Hard