Question

If the slope of the tangent to the curve $y=x^3$ at a point is equal to the ordinate of a point, then the point is:

• A. $\left(27,3\right)$
• B. $\left(3,27\right)$
• C. $\left(1,2\right)$
• D. $\left(-1,3\right)$

Answer 1

The correct option is B

$\left(3,27\right)$

Explanation for the correct option

Given, the equation of the curve $y=x^3$.

To find the slope of the tangent curve differentiate $y=x^3$ with respect to $x$ we get:

$\frac{dy}{dx}=3x^2$

According to the question:

$$\begin{gathered} \Rightarrow 3x^2=y \\ \Rightarrow 3x^2=x^3 \\ \Rightarrow x^3-3x^2=0 \\ \Rightarrow x^2\left(x-3\right)=0 \end{gathered}$$

Using zero property of product we get: $x=0,3$

Thus the points are: $\left(0,0\right)\&\left(3,27\right)$.

Hence, option B is correct.