How To Find Minimum And Maximum Calculator
Tool to decide the maximum value of a role: the maximal value that can take a function. Information technology is a global maximum and non a local maximum.
Maximum of a Function - dCode
Tag(s) : Functions
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Maximum of a Office
- Mathematics
- Functions
- Maximum of a Function
Maximum Computer
Minimum Calculator
Answers to Questions (FAQ)
What is the definition of a maximum of a function?
For whatever 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) $.
Instance: Maximize $ f(ten) = -x^two $, defined over $ \mathbb{R} $, the part reaches its maximum in $ 10=0 $, $ f(x=0) = 0 $ and $ f(x) <= 0 $ over $ \mathbb{R} $.
The maximum of a part is always divers with an interval, it can be local (between two values), or global: over the domain of definition of the role.
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(ten) = -x^2 + 1 $. This role has for derivative $ f'(x) = -2x $ which is nullable in $ x = 0 $ every bit $ f'(x) = 0 \iff -2x = 0 \iff ten = 0 $. An extremum is establish in 0, its value is $ f(0) = ane $. Calculations of limits $$ \lim_{10\to0^-}{f'(x) = 0^+} \\ \lim_{ten\to0^+}{f'(x) = 0^-} $$ show that the derivative change of sign from positive $ 0^+ $ to negative $ 0^- $. Global extremum of the office is then $ 1 $ for $ 10=0 $.
How to calculate a local maximum over an interval?
Add ane or more constraints indicating the conditions for each variable.
Case: 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 farthermost 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 part?
The majorant is any value greater than or equal to the maximum value reached by the function.
What is the maximum of a constant part?
A constant function $ f (x) = c $ is a line, and always equals $ c $, then its maximum is $ c $, reached for any value of $ ten $
What is the maximum of an affine function?
An affine role $ 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 $ 10 $ tends to $ +\infty $
What is the maximum of a 2nd degree polynomial role?
For a quadratic polynomial part $ f(x) = ax^ii + bx + c $ and so
— 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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