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NARRATOR: Exercise seven.

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Let f(x) = x cubed plus x – 2.

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Show that f(x) has
no stationary points.

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So we know that
stationary points occur

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when the derivative is equal to 0.

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So let's first consider
the derivative of this function.

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So f dash x is equal to
3x squared + 1.

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And for stationary points, we need
the derivative to be equal to 0.

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So we have a quadratic equation here.

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So we can think about
the discriminant of this equation.

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And the discriminant we know
is b squared – 4ac

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which in the case of this equation
is 0 squared – 4 times 3 times 1.

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So we have a discriminant of –12.

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So that tells us that f dash x = 0
has no solutions ...

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... since we know that
the discriminant is negative.

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So therefore, if the f dash x = 0
has no solutions,

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that means f(x)
has no stationary points.