Question

Let us consider two points A and B on an arbitrary curve y=f(x). The magnitude of double

derivative (d2y/dx2) is higher at point A. The tangent to the curve at both the points are equal. What we can

conclude about the radius of curvature at point A and B?

1)Radius of curvature at A and B are equal.

2)Radius of curvature at A is higher than that at B.

3)Radius of curvature at A is lower than that at B.

4)Not enough information

Answer 1

Let the curvature be represented by R.

We know that

$R=\left | \frac{(1+(\frac{dy}{dx})^2)^\frac{3}{2}}{\frac{d^2y}{dx^2}} \right |$

Now, we have been given that the the tangent to the curve at both the points A and B are equal. Thus from the formula for R we see that the $\frac{dy}{dx}$ will be a constant thus making the entire numerator to be a constant.

Now,  if the magnitude of the double  derivative $\frac{d^2y}{dx^2}$ is higher at point A, then we can see from the equation for R that the radius of curvature, R will be less at A. This is because $\frac{d^2y}{dx^2}$  comes at the denominator and higher the denominator, lower will be the overall fraction.

Thus, B will have a higher radius of curvature than A. Therefore, out of the given options, option 3 is correct.

Therefore, the answer is:

3)Radius of curvature at A is lower than that at B

Answer 2

Give the ebb and flow a chance to be spoken to by R.

We realize that Now, we have been given that the digression to the bend at both the focuses An and B are equivalent.

In this manner from the equation for R we see that will be a consistent along these lines making the whole numerator to be a steady.