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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
Let $f(x)=x^3+3 x^2-9 x+2$. Then, $f(x)$ has,
  • A
    a maximum at $x = 1$
  • ✓
    a minimum at $x = 1$
  • C
    netither a maximum nor a minimum at $x = -3$
  • D
    none of these.
✓

Answer

Correct option: B.
a minimum at $x = 1$
$f(x)=x^3+3 x^2-9 x+2$
$\Rightarrow f^{\prime}(x)=3 x^2+6 x-9$
local minima or maxima must have $f^{\prime}(x)=0$
$x^3+3 x^2-9 x+2=0$
$x^2+2 x-3=0$
$\Rightarrow(x+3)(x+1)=0$
$\Rightarrow x=-3$ or $x=1$
$f^{\prime \prime}(x)=6 x+6$
$\Rightarrow f^{\prime \prime}(-3)=-12<0$
At, $x=-3$ local maxima.
$f^{\prime \prime}(1)=12>0$
At, $x=1$ local minima.

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