local maximum

Examples of local maximum in the following topics:

• Concavity and the Second Derivative Test

  • The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
  • In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
  • If $f''(x) < 0$ then f(x) has a local maximum at $x$.
  • If $f''(x) > 0$ then f(x) has a local minimum at $x$.
  • Calculate whether a function has a local maximum or minimum at a critical point using the second derivative test

• Maximum and Minimum Values

  • The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
  • The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).
  • The second partial derivative test is a method in multivariable calculus used to determine whether a critical point $(a,b, \cdots )$ of a function $f(x,y, \cdots )$ is a local minimum, maximum, or saddle point.
  • If M(a,b)>0M(a,b)>0 and fxx(a,b)<0f_{xx}(a,b)<0, then $(a,b)$ is a local maximum of $f$.
  • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

• Maximum and Minimum Values

  • A real-valued function $f$ defined on a real line is said to have a local (or relative) maximum point at the point $x_{\text{max}}$, if there exists some $\varepsilon > 0$ such that $f(x_{\text{max}}) \geq f(x)$ when $\left | x - x_{\text{max}} \right | < \varepsilon$.
  • The global maximum and global minimum points are also known as the arg max and arg min: the argument (input) at which the maximum (respectively, minimum) occurs.
  • Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the domain.
  • So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one.
  • One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test or second derivative test.

• Applications of Minima and Maxima in Functions of Two Variables

  • We have learned how to find the minimum and maximum in multivariable functions.
  • In particular, we learned about the second derivative test, which is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum, using the value of the second derivative at the point.
  • at $\left(\frac{3}{8}, -\frac{3}{4}\right)$ $f(x, y)$ has a local maximum, since $f_{xx} = -\frac{3}{8} < 0$
  • Identify steps necessary to find the minimum and maximum in multivariable functions

• Optimization in Several Variables

  • After finding out the function $f(x)$ to be optimized, local maxima or minima at critical points can be easily found.
  • (Of course, end points may have maximum/minimum values as well.)

• Optimization

  • After finding out the function $f(x)$ to be optimized, local maxima or minima at critical points can be easily found.
  • (Of course, end points may have maximum/minimum values as well.)

• Lagrange Multiplers

  • The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints.
  • In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints.
  • Sufficient conditions for a minimum or maximum also exist.
  • Where the Lagrange multiplier $\lambda=0$ we can have a local extremum and the two contours cross instead of meeting tangentially.
  • Therefore where the constraint $g=c$ crosses the contour line $f=-1$, is a local minimum of $f$ on the constraint.

• Applications to Economics and Biology

  • In calculus terms, consumer surplus is the derivative of the definite integral of the demand function with respect to price, from the market price to the maximum reservation price—i.e. the price-intercept of the demand function:

• Planetary Motion According to Kepler and Newton

  • The maximum distance occurs when the angle is 180 degrees.

• Implicit Differentiation

  • Geometrically, the graph defined by $R(x, y) = 0$ will overlap locally with the graph of some equation $y = f(x)$.