Question

The equation of the tangent to the curve$y=x+\frac{4}{x^2}$, that is parallel to the $x$-axis is

• A. $y=0$
• B. $y=1$
• C. $y=2$
• D. $y=3$

Answer 1

The correct option is D

$y=3$

Explanation for the correct options:

First order derivative of the equation representing the curve gives the slope at a point

$y=x+\frac{4}{x^2}=x+4x^{-2}$

Differentiating with respect to $x$ we get

$\Rightarrow$$\frac{dy}{dx}=1+\left(-2\right)4x^{-3}$

$\Rightarrow$$\frac{dy}{dx}=1-\frac{8}{x^3}$

$\Rightarrow$ $\frac{dy}{dx}=m=1-\frac{8}{x^3}$

The tangent is parallel to the $x$-axis.

Hence, the slope of the tangent is $0$ …[Slope of a line parallel to $x$-axis is $0$]

$\Rightarrow$$1-\frac{8}{x^3}=0$

$\Rightarrow$ $x^3=8$

$\Rightarrow$ $x=2$

Substituting this value of $x$ in the equation of the curve we get,

$y=2+\frac{4}{2^2}$

$\Rightarrow$$y=2+\frac{4}{4}$

$\Rightarrow$$y=3$

Hence, the point at which the equation of tangent is required is $\left(2,3\right)$

By using slope point form of equation we get

$y-y_1=m\left(x-x_1\right)$

Substituting all required values we get,

$y-3=0\left(x-2\right)$

$\Rightarrow$ $y=3$

Hence, $y=3$ is the equation of tangent to $y=x+\frac{4}{x^2}$ parallel to $x$-axis

Hence, option $\left(D\right)$ is the correct answer.