Question

i) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the x-axis.

ii) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the y-axis

Answer 1

i) Use implicit differentiation, then set dy dx = 0.

$$\dfrac{d}{dx}(x(x+y^2))=\dfrac{dy}{dx}$$

$$2x+y^2+2xy\dfrac{dy}{dx}=\dfrac{dy}{dx}$$

$$\dfrac{dy}{dx}=\dfrac{2x+y^2}{1-2xy}$$

$$\dfrac{dy}{dx}=0=>\dfrac{2x+y^2}{1-2xy}=0$$

$$2x+y^2=0, 1-2xy\not=0$$

$$x=-\dfrac{y^2}{2}$$

Substitute

$$-\dfrac{y^2}{2}(-\dfrac{y^2}{2}+y^2)=y$$

$$-\dfrac{y^4}{4}=y$$

$y_1=0, x_1=0, Point(0, 0)$

$$y^3=-4$$

$y_2=-\sqrt[3]{4}, x_2=-\sqrt[3]{2}, Point (-\sqrt[3]{2}, -\sqrt[3]{4})$

$Point(0, 0), Point (-\sqrt[3]{2}, -\sqrt[3]{4})$

ii) ) The points with vertical tangents are those where the denominator of $\dfrac{dy}{dx}$ is zero (making the slope undefined). From part (i), we have

$$\dfrac{dy}{dx}=\dfrac{2x+y^2}{1-2xy}=>1-2xy=0$$

$$x=\dfrac{1}{2y}, y\not=0$$

Substitute

$$\dfrac{1}{2y}(\dfrac{1}{2y}+y^2)=y$$

$$\dfrac{1}{4y^2}=\dfrac{y}{2}$$

$$y=\dfrac{\sqrt[3]{4}}{2}$$

$$x=\dfrac{\sqrt[3]{2}}{2}$$

$Point(\dfrac{\sqrt[3]{2}}{2}, \dfrac{\sqrt[3]{4}}{2})$