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Question

# Horizontal Tangent Line Determine the point(s) in the interval $(0,2 \pi)$ at which the graph of $f(x)=2 \cos x+\sin 2 x$ has a horizontal tangent line.

Solution

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Since the slope of the tangent line equals the derivative of the function at the point of tangency, we actually need to find the points that make the derivative equal to zero. This is because if the slope is zero, that means that the tangent of the angle, between the positive part of the $x$-axis and the tangent line, is zero and that occurs if the angle is also zero. Therefore, for such points the tangent line is horizontal.

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