Category: Studying Changes of Function

Condition of Constancy of the Function

Fact. Suppose function $$${y}={f{{\left({x}\right)}}}$$$ is defined and continuous on interval $$${X}$$$ and has finite derivative $$${f{'}}{\left({x}\right)}$$$. Function $$${y}={f{{\left({x}\right)}}}$$$ is constant if and only if $$${f{'}}{\left({x}\right)}$$$ for all $$${x}$$$ in $$${X}$$$.

Condition of Monotony of the Function

Now let's formulate conditions when function will be monotonic, i.e. increasing, decreasing, non-decreasing, non-increasing.

Fact 1. Suppose function $$${y}={f{{\left({x}\right)}}}$$$ is defined and continuous on interval $$${X}$$$ and has finite derivative $$${f{'}}{\left({x}\right)}$$$ on it. Then function is non-decreasing (non-increasing) on $$${X}$$$ if and only if $$${f{'}}{\left({x}\right)}\ge{0}$$$ $$$\left({f{'}}{\left({x}\right)}\le{0}\right)$$$ for all $$${x}$$$ in $$${X}$$$.

Proving Inequalities Using Monotony of the Function

Monotony of the function can be used to prove some not obvious inequalities.

Example 1. Prove that $$${\sin{{\left({x}\right)}}}>\frac{{2}}{\pi}{x}$$$ when $$${0}<{x}<\frac{\pi}{{2}}$$$.

Consider function $$${f{{\left({x}\right)}}}=\frac{{{\sin{{\left({x}\right)}}}}}{{x}}$$$ defined on interval $$${\left({0},\frac{\pi}{{2}}\right]}$$$.

Maxima and Minima. Necessary Conditions

If function $$${y}={f{{\left({x}\right)}}}$$$ is defined and continuous on interval $$${\left[{a},{b}\right]}$$$ is not monotonic, then exist such subintervals $$${\left[\alpha,\beta\right]}$$$ of interval $$${\left[{a},{b}\right]}$$$ that largest and smallest value function takes at inner point, i.e. between $$$\alpha$$$ and $$$\beta$$$.

First Derivative Test

We know that if point is extremum then it is stationary point. So, we need to find all stationary points and test them. But how?

For this we use following test.

First Derivative Test. Suppose that $$${c}$$$ is a stationary point of a continuous function $$${y}={f{{\left({x}\right)}}}$$$.

Second Derivative Test

Instead of using First Derivative Test we can use another test.

Second Derivative Test.

• if $$${f{'}}{\left({c}\right)}={0}$$$ and $$${f{''}}{\left({c}\right)}<{0}$$$ then there is maximum at point $$${c}$$$.
• if $$${f{'}}{\left({c}\right)}={0}$$$ and $$${f{''}}{\left({c}\right)}>{0}$$$ then there is minimum at point $$${c}$$$.
• if $$${f{'}}{\left({c}\right)}={0}$$$ and $$${f{''}}{\left({c}\right)}={0}$$$ then we can't say anything about point $$${c}$$$.

This test is used not so often as first derivative test because of two reasons:

Higher-Order Derivative Test

As we know Second derivative test is inconlusive when for critocal point $$${c}$$$ $$${f{''}}{\left({c}\right)}={0}$$$.

In this case we need to use another test.

Higher-Order Derivative Test. Suppose that $$${k}$$$ $$$\left({k}>{2}\right)$$$ is the smallest number for which $$${{f}}^{{{\left({k}\right)}}}{\left({c}\right)}\ne{0}$$$. If $$${k}$$$ is and odd number, then there is no maximum or minimum at $$${c}$$$. If $$${k}$$$ is even number then $$${c}$$$ is maximum if $$${{f}}^{{{\left({k}\right)}}}{\left({c}\right)}<{0}$$$, and $$${c}$$$ is minimum if $$${{f}}^{{{\left({k}\right)}}}{\left({c}\right)}>{0}$$$.

Closed Interval Method

In fact when we used derivative to find extrema, this extrema mainly were local.

Now we turn our attention to finding global extrema on closed interval $$${\left[{a},{b}\right]}$$$.

Extreme Value Theorem. If $$${f{}}$$$ is continuous on a closed interval $$${\left[{a},{b}\right]}$$$, then $$${f{}}$$$ attains a global maximum value $$${f{{\left({c}\right)}}}$$$ and a global minimum value $$${f{{\left({d}\right)}}}$$$ at some numbers $$${c}$$$ and $$${d}$$$ from interval $$${\left[{a},{b}\right]}$$$.