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The curve y=x^{\frac{1}{5}} has at (0, 0)
A. a vertical tangent (parallel to y-axis)
B. a horizontal tangent (parallel to x-axis)
C. an oblique tangent
D. no tangent

Answers (1)

Given y=x^{\frac{1}{5}}

Differentiate both sides with x and get

\frac{d y}{d x}=\frac{d\left(x^{\frac{1}{5}}\right)}{d x}
Apply power rule and get
\\\Rightarrow \frac{d y}{d x}=\frac{1}{5} \times x^{\frac{1}{5}-1} \\\Rightarrow \frac{d y}{d x}=\frac{1}{5} \times x^{-\frac{4}{5}} \\\Rightarrow \frac{d y}{d x}=\frac{1}{5 x^{\frac{4}{5}}}
Now at (0,0)
\\\Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(0,0)}=\frac{1}{5 \mathrm{x}^{\frac{4}{5}}} \\\Rightarrow\left(\frac{d y}{d x}\right)_{(0,0)}=\frac{1}{5(0)^{\frac{4}{5}}}\\ \Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{(0,0)}=\infty
So the curve \mathrm{y}=\mathrm{x}^{\frac{1}{5}} at (0,0) has vertical tangent parallel to Y-axis.

Hence the correct answer is option A.

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