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Applications of Derivatives — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives2 Marks
Question
Given an interval $[a, b]$ that satisfies hypothesis of Rolle's theorem for the function $f(x)=x^3-2 x^2+3$. It is known that $a=0$. Find the value of $b$.
✓

Answer

Given that $f(x)=x^3-2 x^2+3$
Let $g(x)=x^3-2 x^2=x^2(x-2)$
From (I), $\quad f(x)=g(x)+3$
We see that $g(x)$ becomes zero for $x=0$ and $x=2$.
We observe that for $x=0$,
$
f(0)=g(0)+3=3
$
and for $x=2$,
$
f(2)=g(2)+3=3
$
$\therefore \quad$ We can write that $f(0)=f(2)=3$
It is obvious that the function $f(x)$ is everywhere continuous and differentiable as a cubic polynomial. Consequently, it satisfies all the conditions of Rolle's theorem on the interval $[0,2]$.
So
$
b=2 \text {. }
$

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