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Consider the function $f(x)=\cos(\sin(\cos(x)))$, with $x$ measured in radians.

What turning points can you find?

What are the maximum and minimum values of the function?
Did you know ... ?

This function is bounded, continuous and differentiable at all points. Mathematicians often use knowledge of conditions such as these to deduce lots of information about the properties of functions without the need for extensive calculation. In first year undergraduate analysis courses theorems are rigorously stated and proved which support intuitive statements such as 'between any two maxima a minimum must be found if the function is finite, continuous and differentiable'.