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Minimum of a Function

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

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Minimum of a Function -

Tag(s) : Functions

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# Minimum of a Function

## Minimum Calculator

 Search Domain Real Numbers (R) Integers (Z)

## Maximum Calculator

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

For any function $f$ defined on an interval $I$ and $m$ a real number belonging to $I$, if $f(x) <= f(m)$ on the interval $I$ then $f$ reaches its minimum in $x=m$ over $I$. In that case, $f(m)$ is the minimum value of the function, reached when $x=m$.

The minimum of a function is always defined with an interval (which may be the domain of definition of the function).

### How to find the minimum of a function?

Finding the minimum of a function $f$, is equivalent to calculate $f(m)$. To find $m$, use the derivative of the function. The minimum value of a function is found when its derivative is null and changes of sign, from negative to positive.

Example: $f(x) = x^2$ defined over $\mathbb{R}$, its derivative is $f'(x) = 2x$, that is equal to zero in $x = 0$ because $f'(x) = 0 \iff 2x = 0 \iff x=0$. The derivative goes from negative to positive in $x = 0$ so the function has a minimum in $x=0$, $f(x=0) = 0$ and $f(x) >= 0$ over $\mathbb{R}$.

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

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

Example: Find the minimum of $\sin{x}$ for $0 < x < \pi$

Indicate several equations with the operator logical AND && to separate the equations

### What is an extremum?

An extremum is an extreme value of a function, this value can be maximum (the maximum value of the function) or minimum (the minimum value of the function).

### What is a minorant of a function?

The minorant is any value lower than or equal to the minimum value reached by the function.

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

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

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

An line/affine function $f (x) = ax + b$ always has for minimum $-\infty$

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

- If $a > 0$, the minimum of $f$ is $-\infty$ when $x$ tends to $-\infty$

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

A quadratic polynomial function of the form $f (x) = ax ^ 2 + bx + c$ then

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

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

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