Applications of Derivatives — Maths STD 12 Science — Question
Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives3 Marks
Question
Verify Rolle's theorem for the function$f(x)=x^2-4 x+10 \text { on }[0,4] .$
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Answer
Given that $f(x)=x^2-4 x+10$ $f(x)$ is a polynomial which is continuous on $[0,4]$ and it is differentiable on $(0,4)$. Let $a=0$, and $b=4$ For $x=a=0$ from (I) we get, $ f(a)=f(0)=(0)^2-4(0)+10=10 $ For $x=b=4$ from (I) we get, $ f(b)=f(4)=(4)^2-4(4)+10=10 $ So, here $f(a)=f(b) \quad$ i.e. $f(0)=f(4)=10$ All the conditions of Rolle's theorem are satisfied. To get the value of $c$, we should have $f^{\prime}(c)=0$ for some $c \in(0,4)$ Differentiate (I) w.r.t.x. $f^{\prime}(x)=2 x-4=2(x-4)$ Now, for $x=c$, $f^{\prime}(c)=0 \Rightarrow 2(c-2)=0 \Rightarrow c=2$ Also $c=2 \in(0,4)$ Thus Rolle's theorem is verified.
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