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Application of Derivatives — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsApplication of Derivatives1 Mark
MCQ
The function $f(x)=x^5-5 x^4+5 x^3-1$ has
  • A
    one minima and two maxima
  • B
    two minima and one maxima
  • C
    two minima and two maxima
  • ✓
    one minima and one maxima
✓

Answer

Correct option: D.
one minima and one maxima
(d) : Given, $f(x)=x^5-5 x^4+5 x^3-1$
Then, $f^{\prime}(x)=5 x^4-20 x^3+15 x^2$
$
\Rightarrow f^{\prime \prime}(x)=20 x^3-60 x^2+30 x
$
Now, $f^{\prime}(x)=0 \Rightarrow 5 x^2\left(x^2-4 x+3\right)=0$
$
\begin{array}{l}
\Rightarrow \quad x=0,1,3 \\
f^{\prime \prime}(1)=-10<0 \text { and } f^{\prime \prime}(3)=90>0 \\
f^{\prime \prime}(0)=0 \text { and } f^{\prime \prime \prime}(0) \neq 0
\end{array}
$
So, $x=0$ is a point of inflexion.
$\therefore f(x)$ has maximum at $x=1$ and minimum at $x=3$.

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