AP EAMCET · Maths · Continuity and Differentiability
Let \(f(x)\) be a differentiable function such that \(f(1)=2\), \(f(2)=6\) and \(f(x+y)=f(x)+k x y+\frac{4}{3} y^2 \forall x, y \in \mathbb{R}\) then \(f(x)=\)
- A \(4 x-2\)
- B \(y-4 x^2+2 x-4\)
- C \(\frac{8}{3} x^2+\frac{4}{3}\)
- D \(\frac{4}{3} x^2+\frac{2}{3}\)
Answer & Solution
Correct Answer
(D) \(\frac{4}{3} x^2+\frac{2}{3}\)
Step-by-step Solution
Detailed explanation
\(f(1)=2, f(2)=6\) \(f(x+y)=f(x)+k x y+\frac{4}{3} y^2... (I)\) Put \(x=1 \& y=1\) \(\begin{aligned} & \therefore f(1+1)=f(1)+k+\frac{4}{3} \Rightarrow 6-\frac{4}{3}=2+k \\ & \Rightarrow k=\frac{8}{3} \end{aligned}\) Put \(x=1\) and \(y=m-1\) in equation (i), we get…
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
- The coefficient of the highest power of \(x\) in the expansion of \(\left(x+\sqrt{x^2-1}\right)^8+\left(x-\sqrt{x^2-1}\right)^8\) isAP EAMCET 2023 Hard
- The number of integers greater than 6000 that can be formed by using the digits \(0,5,6,7,8\) and 9 without repetition isAP EAMCET 2025 Medium
- If \(f(x)=x e^{x(1-x)}, x \in \mathbb{R}\), then \(f(x)\) isAP EAMCET 2025 Medium
- The direction cosines of two lines are \(\left\langle\frac{\sqrt{3}}{2}, \frac{1}{4}, \frac{\sqrt{3}}{4}\right\rangle\) and \(\left\langle\frac{-\sqrt{3}}{2}, \frac{1}{4}, \frac{\sqrt{3}}{4}\right\rangle\). Then the angle between the lines is equal toAP EAMCET 2020 Easy
- If \(y=x \operatorname{Tan}^{-1}\left(\frac{x}{y}\right)\), then \(\frac{d y}{d x}=\)AP EAMCET 2017 Medium
- Let origin be the centre, be the foci and be the eccentricity of a hyperbola. Then the lineAP EAMCET 2022 Medium
More PYQs from AP EAMCET
- What is the nature of reaction at , if the entropy change and enthalpy change for a chemical reaction are and, respectively.AP EAMCET 2019 Easy
- The molecular formula of potash alum isAP EAMCET 2004 Easy
- If \(P\) divides the line segment joining the points \(A\) and \(B\) in the ratio \(2: 1\) and the position vectors of \(A\) and \(B\) are \(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}\) and \(-3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}\) respectively, then the position vector of \(p\) isAP EAMCET 2020 Easy
- The equilibrium concentrations of \(\mathrm{N}_2, \mathrm{H}_2\) and \(\mathrm{NH}_3\) in the formation of \(\mathrm{NH}_3\) at \(500 \mathrm{~K}\) are \(1.25 \times 10^{-2} \mathrm{M}, 4.0 \times 10^{-2} \mathrm{M}\) and \(1.6 \times 10^{-2} \mathrm{M}\) respectively. The equilibrium constant \(\mathrm{K}_{\mathrm{p}}\) at the same temperature isAP EAMCET 2018 Easy
- If m and M are the absolute minimum and absolute maximum values of the function \(\mathrm{f}(\mathrm{x})=2 \sqrt{2} \sin \mathrm{x}-\tan \mathrm{x}\) in the interval \([0, \pi / 3]\), then \(\mathrm{m}+\mathrm{M}=\)AP EAMCET 2025 Medium
- A system is taken from state to state along two different paths. The heat absorbed and work done by the system along these two paths are and respectively, thenAP EAMCET 2021 Medium