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Question

Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

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Solution

The equation of the given curve is y = x3.

The slope of the tangent at the point (x, y) is given by,

When the slope of the tangent is equal to the y-coordinate of the point, then y = 3x2.

Also, we have y = x3.

∴3x2 = x3

x2 (x − 3) = 0

x = 0, x = 3

When x = 0, then y = 0 and when x = 3, then y = 3(3)2 = 27.

Hence, the required points are (0, 0) and (3, 27).


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