Examples of local maximum in the following topics:
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- 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
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- 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
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- Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.
- The graph attains a local maximum at $(1,2)$ because it is the highest point in an open interval around $x=1$.
- The local maximum is the y-coordinate at $x=1$ which is $2$.
- For the function pictured, the local maximum is at the $y$-value of 16, and it occurs when $x=-2$.
- This graph has examples of all four possibilities: relative (local) maximum and minimum, and global maximum and minimum.
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- 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.
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- 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
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- 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.)
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- 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.
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- Third, Title I provided standards of maximum work hours, minimum wages, and labor conditions that the codes would cover.
- It also provided funding for a series of transportation projects, local initiatives that would battle unemployment through public work projects, and necessary acquisitions of property that would make such projects possible.
- Minimum wages, maximum working hours, prices, and production quotas were all to be covered under the codes.
- Johnson called on every business establishment in the nation to accept a stopgap "blanket code": a minimum wage of between 20 and 45 cents per hour, a maximum workweek of 35–45 hours, and the abolition of child labor.
- Between 4,000 and 5,000 business practices were prohibited, some 3,000 administrative orders running to over 10,000 pages promulgated, and thousands of opinions and guides from national, regional, and local code boards interpreted and enforced the Act.
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- The U.S. federal, state and local governments levy taxes on individuals based on income, property, estate transfers, and/or sales transactions.
- Taxes based on income are imposed at the federal, most state, and some local levels.
- Gifts of money or property to qualifying charitable organizations, subject to certain maximum limitations
- State income tax rates vary from 1% to 16%, including local income tax where applicable.
- Many cities, counties, transit authorities, and special purpose districts impose an additional local sales tax.
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- Likewise, a buyer has a certain maximum price at which s/he is willing to buy, though s/he would happily pay less.
- If the minimum the seller would accept is less than the maximum a buyer would pay, a transaction can occur.
- There are markets for many types of products other than stocks: the global oil market, your local farmers' market, and eBay are all forms of markets with their own defining characteristics.