local minimum

Examples of local minimum in the following topics:

• 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.
  • If $M(a,b)>0$ and $f_{xx}(a,b)>0$, then $(a,b)$ is a local minimum of $f$.
  • For example, if a bounded differentiable function $f$ defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem).
  • Its only critical point is at $(0,0)$, which is a local minimum with $f(0,0) = 0$.
  • Apply the second partial derivative test to determine whether a critical point is a local minimum, maximum, or saddle point

• 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 minimum at $x$.
  • Now, by the first derivative test, $f(x)$ 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

  • Similarly, a function has a local minimum point at $x_{\text{min}}$, if $f(x_{\text{min}}) \leq f(x)$ when $\left | x - x_{\text{min}} \right | < \varepsilon$.
  • 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

• Lagrange Multiplers

  • Sufficient conditions for a minimum or maximum also exist.
  • Every point $\left(\frac{-\pi}{2}, y\right)$$f=-1$ is a global minimum of $f$ with value $-1$.
  • Therefore where the constraint $g=c$ crosses the contour line $f=-1$, is a local minimum of $f$ on the constraint.
  • The trace and the contour $f=-1$ cross at the minimum as we can see in the figure.
  • Since both $g_x \neq 0$ and $g_y \neq 0$, the Lagrange multiplier $\lambda = 0$ at the minimum.

• 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.)

• Planetary Motion According to Kepler and Newton

  • The minimum distance occurs when the angle is 0.

• Implicit Differentiation

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

• The Mean Value Theorem, Rolle's Theorem, and Monotonicity

  • The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.