Applications of Derivatives — Maths STD 12 Science — Question
Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives3 Marks
Question
Verify Lagrange's mean value theorem for the function $f(x)=x+\frac{1}{x}$ on the interval $[1,3]$.
✓
Answer
Given that $f(x)=x+\frac{1}{x}$ The function $f(x)$ is continuous on the closed interval $[1,3]$ and differentiable on the open interval $(1,3)$, so the LMVT is applicable to the function. Differentiate (I) w. r.t. $x$. $f^{\prime}(x)=1-\frac{1}{x^2}$ Let $a=1$ and $b=3$ From (I), $ \begin{aligned} & f(a)=f(1)=1+\frac{1}{1}=2 \\ & f(b)=f(3)=3+\frac{1}{3}=\frac{10}{3} \end{aligned} $ Let $c \in(1,3)$ such that $ \begin{aligned} & f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} \\ & 1-\frac{1}{c^2}=\frac{\frac{10}{3}-2}{3-1} \\ & 1-\frac{1}{c^2} \quad= \frac{\frac{4}{3}}{2}=\frac{2}{3} \\ & \therefore \quad c^2=3 \Rightarrow c= \pm \sqrt{3} \\ & \therefore \quad c=\sqrt{3} \in(1,3) \text { and } c=-\sqrt{3} \notin(1,3) \end{aligned} $
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.