Calculus homework help?!?
The function f(x) = -2 x^3 + 21 x^2 - 36 x + 5 has one local minimum and one local maximum.
This function has a local minimum at x equals ? with value ?
and a local maximum at x equals ? with value ?
A local extremum is a point on a graph where the function reaches either a maximum or minimum value, and it is surrounded by values that are either higher or lower than itself.
To find local extrema on a graph, you can take the derivative of the function and set it equal to zero. The values of x where the derivative equals zero are potential local extrema. You can then use the second derivative test to determine if these points are local maxima or minima.
A local extremum is a point where the function reaches a maximum or minimum within a specific interval, while a global extremum is the overall maximum or minimum of the entire function. A global extremum can occur at a local extremum, but not all local extrema are global.
Yes, a function can have multiple local extrema. A function can have both local maxima and minima, and there can be multiple of each type.
Local extrema are important in real-world applications because they represent critical points in a function. For example, in economics and finance, local extrema can represent the highest and lowest points of a stock's price. In physics, local extrema can represent the maximum or minimum values of a physical quantity. Identifying and analyzing local extrema can provide valuable insights and help make informed decisions.