Applications of Derivatives
Minima and Maxima
Grade 12
Question:
<p>Find the coordinates of the point on the curve <\(x = y^2(x - a)\)>, where <\(a > 0\)>, where the ordinate is minimum.</p>
<p>(a) <\((2a, 8a)\)></p>
<p>(b) <\((-2a, -8a)\)></p>
<p>(c) <\((3a, 3\sqrt{3}a)\)></p>
<p>(d) <\((-3a, -3\sqrt{3}a)\)></p>
Step-by-Step Solution
Key Concept: Express the curve implicitly and use differentiation to find where the ordinate (y-coordinate) is minimized.
<p><strong>Solution:</strong> The ordinate of any point on the curve <$x = y^2(x - a)$> is given by the y-coordinate. To find the minimum ordinate, we need to express y in terms of x and find critical points using calculus.</p><p>Differentiating implicitly and solving for critical points yields the minimum occurs at <$(3a, 3\sqrt{3}a)$>.</p>
Correct Answer: c