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Maximum of a Function

Tool to determine the maximum value of a function: the maximal value that can take a function. It is a global maximum and not a local maximum.

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Maximum of a Function -

Tag(s) : Functions

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# Maximum of a Function

## Maximum Calculator

 Search Domain Real Numbers (R) Integers (Z)

## Minimum Calculator

### What is the definition of a maximum of a function?

For any function $f$ defined on an interval $I$, taking $m$ a real of this interval, if $f(x) <= f(m)$ over the whole interval $I$ then $f$ reaches its maximum in $x=m$ over $I$. The value of the maximum is $f(m)$.

Example: Maximize $f(x) = -x^2$, defined over $\mathbb{R}$, the function reaches its maximum in $x=0$, $f(x=0) = 0$ and $f(x) <= 0$ over $\mathbb{R}$.

The maximum of a function is always defined with an interval, it can be local (between 2 values), or global: over the domain of definition of the function.

### How to calculate a maximum of a function?

The maximums of a function are detected when the derivative becomes null and changes its sign (passing through 0 from the positive side to the negative side).

Example: Calculate the maximum of the function $f(x) = -x^2 + 1$. This function has for derivative $f'(x) = -2x$ which is nullable in $x = 0$ as $f'(x) = 0 \iff -2x = 0 \iff x = 0$. An extremum is found in 0, its value is $f(0) = 1$. Calculations of limits $$\lim_{x\to0^-}{f'(x) = 0^+} \\ \lim_{x\to0^+}{f'(x) = 0^-}$$ show that the derivative change of sign from positive $0^+$ to negative $0^-$. Global extremum of the function is then $1$ for $x=0$.

dCode has also a minimum of a function calculator tool.

### How to calculate a local maximum over an interval?

Add one or more constraints indicating the conditions for each variable.

Example: Find the maximum of $\cos{x}$ for $-\pi < x < \pi$

Indicate to dCode several equations with the operator && (logical AND) to separate the equations

### What is an extremum?

An extremum is the name given to an extreme value of a function, a value that can be maximum (maximum of a function) or minimal (minimum of a function).

### What is a majorant of a function?

The majorant is any value greater than or equal to the maximum value reached by the function.

### What is the maximum of a constant function?

A constant function $f (x) = c$ is a line, and always equals $c$, so its maximum is $c$, reached for any value of $x$

### What is the maximum of an affine function?

An affine function $f (x) = ax + b$ is a line that always has for maximum $+\infty$

— If $a < 0$, the maximum of $f$ is $+\infty$ when $x$ tends to $-\infty$

— If $a > 0$, the maximum of $f$ is $+\infty$ when $x$ tends to $+\infty$

### What is the maximum of a 2nd degree polynomial function?

For a quadratic polynomial function $f(x) = ax^2 + bx + c$ then

— If $a < 0$, the maximum of $f$ is $(-b^2 + 4 a c)/(4 a)$ reached when $x = -\frac{b}{2a}$

— If $a > 0$, the maximum of $f$ is $+\infty$ when $x$ tends to $+\infty$

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