The curve y=x^ 1 5 has at (0, 0) - Vidyadip

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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