Mathematics - Application of Derivative Question with Solution | TestHub

Let $\left(x_0,x_0^4\right)$ be the point of tangency. Then the equation of the tangent will be $y-x_0^4=y^{\prime }\left(x_0\right)\left(x-x_0\right)$. Since this tangent passes through the point $(2,0)$, we have $0-x_0^4=4x_0^3\left(2-x_0\right)$, or $3x_0^4-8x_0^3=0$. That is, $x_0=0$ or $x_0=\frac{8}{3}$, so that the points of tangency are $(0,0)$ and (8/3, 4096/81). Therefore, the equations of the tangents are $y=0$ and $y-\frac{4096}{81}=\frac{2048}{27}\left(x-\frac{8}{3}\right)$