why we take derivative of a function to reach local or global optimum

Question

I am not able to understand why we take
derivative of a function to check its global or local optimum.
what information does derivative of a function gives us,
that helps us to reach to a global optimum.
I know that derivative of a function is the rate of change of a function with
respect to independent variable
but how that rate of change helps us in determining the global optimum.
Please help with any example.

Answer 1

Given a function $f:ℝ→ℝ$. The derivative in a point $x_0$, $$f'(x_0)$$ can be seen as the slope of the tangent at $f(x_0)$.

Have a look at that great gif from the wikipedia article "derivative".

As the tangent moves from $x=-2$ to $x=-1$ you can see a change in the sign. First it is negative, as the slope is pointing "downwards", then it is positive , as the slope is "pointing upwards".

Note that if the function is "smooth enough" that slope will also be smooth. That means there is no jump from a negative slope to a positive slope. So there will be a point $x^\star$, with $f'(x^\star)=0$. ¹ At that point there is a local minima or maxima.

There is also another case where the slope is positive, gets close to $0$, but then increases again. So $f'$ just "touches" the x-axis. That can be seen in the gif at $x=0$. Such a behavior is called saddle-point.

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¹ $f=|x|$ is an example of a function where such a jump can be seen. I hope to not confuse with this additional information. The slope for $x<0$ is negative and for $x>0$ positive: $$f'(x)=\begin{cases} -1 & x<0 \\ 1 & x>0 \end{cases}$$ But at $x=0$ the slope has to make a sudden jump from negative to positive. In fact $f=|x|$ is not differentiable in $x=0$.

Answer 2

The zeroes of the derivative give you the location of local maxima.

The global maximum can be at one of

• an endpoint of the definition interval,

• a discontinuity point,

• a local maximum.

You must take the point (if unique) among these that achieves the largest function value.