How many stationary points does $y = 2x^3 - 6x^2 + 5x - 7$ have?

How many stationary points does $y = 2x^3 - 6x^2 + 5x - 7$ have?

Question

The function $y = 2x^3 - 6x^2 + 5x - 7$ has

1. no stationary points;

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1. one stationary point;

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1. two stationary points;

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1. three stationary points.

Solution

Differentiating, we find $y' = 6x^2 - 12x + 5.$

We could find the roots, but it is quicker just to notice that the discriminant of the quadratic $y'$ is $12^2 - 4\times 6\times 5 = 24 > 0$.

This means the equation $y' = 0$ has two distinct real roots, and hence the answer is (c).