Question

Find the point on the curve $y={{x}^{3}}-11x+5$ at which the equation of the tangent is $y=x-11$ .

Answer 1

Hint: In order to solve this problem, we need to compare the equation of the line with $y=mx+c$ and with the value of the slope. We can also find the slope with the help of differentiating the equation of the curve with respect to x.

Complete step by step answer:
We are given the equation of the curve and we need to find the point on that curve on which the tangent gives the equation of line as $y=x-11$ .
Lets first find the slope of that line by comparing the equation with $y=mx+c$ , where m is the slope of the line.
Now, by comparing we can see that the slope of the line is 1.
We can also find the slope of the line by differentiation of the equation of the curve with respect to x.
For differentiation, we need to use the following standard formula,
$\dfrac{d{{x}^{n}}}{dx}=n{{x}^{n-1}}$
Let's differentiate the equation of y with respect of x, we get,
$\begin{align}
  & y={{x}^{3}}-11x+5 \\
 & \dfrac{dy}{dx}=3{{x}^{2-1}}-11 \\
\end{align}$
By we know that the value of $\dfrac{dy}{dx}$ is 1, hence substituting we get,
$1=3{{x}^{2}}-11$
Solving this equation for the value of x, we get,
$\begin{align}
  & 1=3{{x}^{2}}-11 \\
 & 3{{x}^{2}}=12 \\
 & {{x}^{2}}=4 \\
\end{align}$
Taking the square root on both sides we get,
$x=\pm 2$
Substituting the value of x in the equation of the curve we get,
When x = 2,
$\begin{align}
  & y={{2}^{3}}-11\left( 2 \right)+5 \\
 & =8-22+5 \\
 & =-9
\end{align}$
When x = -2,
$\begin{align}
  & y={{\left( -2 \right)}^{3}}-11\left( -2 \right)+5 \\
 & =-8+22+5 \\
 & =19
\end{align}$
Therefore, the coordinate of the point is (2, -9) and (-2,19).
But these points also satisfy the equation of the line.
So, substituting the first point the equation to cross-check, we get,
-9 = 2 – 11 .
Therefore, the point (2,-9) satisfies the equation.
Let's check the second point (-2,19).
$19\ne -2-11$
Hence, we can see that the second point does not satisfy the equation of the line.
Hence, the required point is (2,-9).

Note:
In this question, we get two points that satisfy the equation of the tangent, but according to the definition of the tangent cuts the surface at only one point so only one point will satisfy the equation of the line. So, in order to cross-check we need to substitute both the point and verify which point satisfies the equation.