Question

The curve y = x^1/5 has at (0, 0)
(a) a vertical tangent (b) a horizontal tangent
(c) an oblique tangent (d) no tangent

Answer 1

The given curve is

$y=x^{\frac{1}{5}}$

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=\frac{1}{5}{x}^{-\frac{4}{5}}$

At (0, 0), we have

${\left(\frac{dy}{dx}\right)}_{\left(0,0\right)}=\frac{1}{5}\times {\left(0\right)}^{-\frac{4}{5}}=\frac{1}{5}\times \infty =\infty$

We know that, if $\frac{dy}{dx}=\infty$, then the tangent is parallel to the y-axis.

Now, ${\left(\frac{dy}{dx}\right)}_{\left(0,0\right)}=\infty$

So, the tangent to the given curve at (0, 0) is parallel to the y-axis.

Thus, the curve $y=x^{\frac{1}{5}}$ has at (0, 0) a vertical tangent.

Hence, the correct answer is option (a).