The equation of the tangent to the curve \( y = x + \frac{4}{x^2} \), that is parallel to the x-axis, is?
Options:
\( y = 0 \)
\( y = 1 \)
Answer (2)
The equation of the tangent to the curve y = x + x 2 4 that is parallel to the x-axis is y = 3 , found by determining where the slope of the curve is zero. This occurs at the point (2, 3). The given options do not include the correct answer.
;
To find the equation of the tangent to the curve y = x + x 2 4 that is parallel to the x-axis, we need to determine when the slope of the tangent line is zero. A tangent line that is parallel to the x-axis will have a slope of zero.
First, we need to compute the derivative of the function y = x + x 2 4 .
The derivative y ′ with respect to x is given by:
y ′ = d x d ( x + x 2 4 ) = d x d ( x ) + d x d ( x 2 4 ) .
The derivative of x is 1 , and
The derivative of x 2 4 can be found using the power rule. Rewrite x 2 4 as 4 x − 2 . The derivative of 4 x − 2 is:
d x d ( 4 x − 2 ) = 4 ⋅ ( − 2 ) x − 3 = − x 3 8 .
So the derivative y ′ is:
y ′ = 1 − x 3 8 .
We set this derivative equal to zero to find when the slope of the tangent is zero:
1 − x 3 8 = 0.
Solving for x , we have:
1 = x 3 8 , x 3 = 8 , x = 2.
Now, we substitute x = 2 back into the original equation to find the corresponding y -value:
y = 2 + 2 2 4 = 2 + 1 = 3.
The point at which the tangent is parallel to the x-axis is ( 2 , 3 ) . Therefore, the equation of the tangent line, which is a horizontal line at this point, is:
y = 3.
However, the choices given are y = 0 and y = 1 . Since the computed equation does not match the options provided, the correct answer based on these choices would not be applicable in the context. It's important to properly interpret the settings or reconsider given choices when resolved equations have such discrepancies.
Ensure that calculations are verified according to the context of provided options.
