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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES1 Mark
MCQ
The curve $\text{y}=\text{x}^{\frac{1}{5}}$ has at $(0, 0)$
  • ✓
    A vertical tangent $($parallel to $y-$ axis$)$.
  • B
    A horizontal tangent $($parallel to $x-$ axis$).$
  • C
    An oblique tangent.
  • D
    No tangen.
✓

Answer

Correct option: A.
A vertical tangent $($parallel to $y-$ axis$)$.
We are given that $\text{y}=\text{x}^{\frac{1}{5}}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{5}\text{x}^{\frac{1}{3}-1}\ \Big[\because\frac{\text{d}}{\text{dx}}(\text{x}^\text{n})=\text{nx}^{\text{n}-1}\Big]$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{5}\text{x}^{\frac{-4}{5}}$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{5\text{x}^{\frac{4}{5}}}$
$\Rightarrow\ \Big(\frac{\text{dy}}{\text{dx}}\Big)_{(0,0)}=\frac{1}{5(0)^{\frac{4}{5}}}=\infty$
So, the curve $\text{y}=\text{x}^{\frac{1}{5}}$ has a vertical tangent at $(0, 0),$ which is parallel to $Y-$ axis.

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