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Applications of Derivatives — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives2 Marks
Question
Verify Rolle's theorem for the function $f(x)=x^2-5 x+9$ on $[1,4]$
✓

Answer

The function f given as $f(x)=x^2-5x+9$ is a polynomial function.
Hence
$(i)$ it is continuous on$ [1,4]$
$(ii)$ differentiable on $(1,4).$
Now, $f(1)=1^2-5(1)+9=1-5+9=5$
and $f(4)=4^2-5(4)+9=16-20+9=5$
$f (1)=f(4)$
Thus, the function f satisfies all the conditions of the Rolle’s theorem.
therefore there exists $c \in (1, 4) $such that $f '(c)= 0$
Now, $f(x)=x^2-5 x+9$
$\therefore f^{\prime}(x)=\frac{d}{d x}\left(x^2-5 x+9\right)=2 x-5 \times 1+0$
$=2x-5$
$f'(c)=2c-5$
$f^{\prime}(c)=0$ gives, $2 c-5=0$
$c=\frac{5}{2} \in(1,4)$
Hence, the Rolle’s theorem is verified

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