Given: a particle moves along the curve y = \((\frac{2}{3})x^3+1.\)
To find the points on the curve at which the y - coordinate is changing twice as fast as the x - coordinate.
Equation of curve is y = \((\frac{2}{3})x^3+1\)
Differentiating the above equation with respect to t, we get
When y - coordinate is changing twice as fast as the x - coordinate, i.e.,
Equating equation (i) and equation (ii), we get
Hence the points on the curve at which the y - coordinate changes twice as fast as the x - coordinate are \((1, \frac{5}{3})\) and \((-1, \frac{1}{3})\)
