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For example, if I would like to determine the maximum and the minimum of $ f(x)=\tan(x)- \dfrac{2}{3}x^2-x $ on the interval of $x∈\bigg[-\dfrac{1}{10},\dfrac{1}{10}\bigg]$, is there an equation or something similar to do that? (without drawing the graph of the function). I am looking for a method which can be applied to any function not just to that one.

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    If the function is differentiable (twice makes it even easier), then the answer is yes. (As long as you're working over a compact domain). – 2017-01-28
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    the Maximum in the given Intervall is zero – 2017-01-28
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    The tag functional analysis has nothing to do with the question! – 2017-01-28

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If you have a differentiable function $f(x)$ in an interval $(a,b)$ you can do the following:

  • Find the relative extrema on this interval. Suppose there are in this particular case three of them located at $c_1, c_2, c_3$. Evaluate $f(c_1), f(c_2)$ and $f(c_3)$.
  • Evaluate the function at the limits of the interval: $f(a), f(b)$.

The greatest of those values is the absolute maximum of the function in the interval, and the lowest is the absolute minimum of the function in the interval as well.

Note: to find the extrema (minima and maxima) you can use the fact that at these points the derivative $f'(x)$ is null (impose $f'(x) = 0$ and find the results of this equation)

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    if you use the first derivative you will get $$f'(x)=1+\tan(x)^2-\frac{4}{3}x-1$$ you can not found any solution in an explicit form – 2017-01-28
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    I see your point, but does that invalidate the method? It still is a general method to find a function's maximum and minimum on an interval. This would be another discussion about how to solve the equation obtained by $f'(x) = 0$ – 2017-01-28
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    yes that is what i meant the method is clear but can be non trivial in a special case – 2017-01-28