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Applications of Derivatives — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsApplications of Derivatives2 Marks
MCQ
The function $f(x)=x^3-6 x^2+9 x+2$ has maximum value when $x$ is
  • ✓
    $1$
  • B
    $2$
  • C
    $3$
  • D
    $6$
✓

Answer

Correct option: A.
$1$
Given, $f(x)=x^3-6 x^2+9 x+2$
$\Rightarrow f^{\prime}(x)=3 x^2-12 x+9$
For critical points we must have, $f^{\prime}(x)=0$
$\Rightarrow 3 x^2-12 x+9=0$
$\Rightarrow 3 x^2-3 x-9 x+9=0$
$\Rightarrow(3 x-9)(x-1)=0 $
$\Rightarrow x=3, x=1$
Now, $f^{\prime \prime}(x)=6 x-12$
and $f^{\prime \prime}(3)=18-12=6>0$ and $f^{\prime \prime}(1)=6-12=-6<0$
$\Rightarrow x=1$ is the point of maxima.

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