Question

Let $f\left(x\right)=2x^3+3x\forall x\in R$, then equation of tangent for $y=f^{-1}\left(x\right)$ at $x=5$ will be

• A. $9y-x=4$
• B. $9y-4x=-19$
• C. $49y-9x=4$
• D. $9y-2x=-1$

Answer 1

The correct option is A $9y-x=4$

$y=2x^3+3x$ ____________ (1)

slope $x$ and $y$ and solve for $y$

$x=2y^3+3y$ ___________ (2)

Now take derivate of above equation

$\frac{dx}{dy}=6y^2\frac{dy}{dx}+3.\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{1}{6y^2+3}$ ______ (3)

Use equation (2) to find $y$ at $x=5$

$2y^3+3y=5$

By inspection $y=1$ satisfies the above equation

$\therefore \frac{dy}{dx}=\frac{1}{9}$ ($\because$ slope of tangent) at (5,1)

$(y-1)=\frac{1}{a}(x-5)$

$9y-9=x-5$

$9y-x=4$ is equation of tangent.