A car is moving along the curve \(y = x^3 + 12\). The point(s) on the curve at which the rate of change of its \(y\)-coordinate at a certain time is 3 times the rate of change of its \(x\)-coordinate is/are
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (1, 13) and (−1, 11)
Both x and y change with time t. Differentiating \(y = x^3 + 12\) with respect to t by the chain rule:
\( \displaystyle \frac{dy}{dt} = 3x^2\,\frac{dx}{dt} \)
The condition is \(\frac{dy}{dt} = 3\,\frac{dx}{dt}\), so (with \(\frac{dx}{dt} \ne 0\)):
\(3x^2 = 3\)
\(x^2 = 1,\quad x = \pm 1\)
The y-coordinates from the curve:
\(x = 1:\ y = 1 + 12 = 13\)
\(x = -1:\ y = -1 + 12 = 11\)
So the points are (1, 13) and (−1, 11).