# Questions tagged [maxima-minima]

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire ...

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### What can you do if the higher-order derivative test is inconclusive?

The second-derivative test states that if $x$ is a real number such that $f'(x)=0$, then: If $f''(x)>0$, then $f$ has a local minimum at $x$. If $f''(x)<0$, then $f$ has a local maximum at $x$...
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### Integration of a function that is the extreme value of a function having three variables

Consider $$\phi(a,b,t) = a^4 - 5a^2 +b^2 + 5t^2 -4bt -2t + \frac{33}{4}$$ where $a,b,t ∈ R$.Given that $f(t)$ and $g(b)$ are the minimum values of $\phi(a,b,t)$. Based on this statement there are two ...
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### properties of $\min(x_1…x_n)$

I want to take measurements of algorithm performance. I have two algorithms A and B that run one after the other (composition) I want to measure how well the composition of algorithms is better than ...
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### Any way to mathematically express the set of all argmin(f(x))?

my question is: I would like to express in a mathematical way the first $argmin(f(x))$. The function $argmin$ returns the argument for the global minima, but, there is any way to express the set of ...
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### An absolute minima can be considered local?

local minimum value of f if $f(c) \leqslant f(x)$ when x is near c. I have reading comprehension problem... By the definition above can an absolute minimun also be included as local?
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### Deriative of ${\frac{(\ln x)^{2}}{\sqrt{x}}}$ is not correct [closed]

Using the chain and quotient rules, I get: $${\frac{2*\ln x}{x*\sqrt{x}}-\frac{(\ln x)^{2}}{2*\sqrt{x}}}$$ Answer:$(0,0)$ - min, $(1,0)$- min, $(e^4$,$\frac{16}{e^2})$ - max Can anyone please ...
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### Maximizing the value of a two variable function along any curve

I read that, of all the points on an origin-centered circle in the x-y plane, the function $z=ax+by$ is maximum (or minimum) at the point where $\frac{x}{y}=\frac{a}{b}$ I think this is too specific. ...
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### Ways to find $\min[f(x)]$ where $f(x) = (x-1)(x-2)(x-3)(x-4)$ without using derivatives

As title states I need to: Find $\min[f(x)]$ where $f(x) = (x-1)(x-2)(x-3)(x-4)$ without using derivatives Since I i'm restricted to not use the derivatives I've started to play with the function ...
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### How do i know $x(t)$ must be the largest or smallest value when I let $x'(0)=0$,and substitute the $s_k$ value back into this formula?

How do i know when i let $x'(0)=0$,x(t) must have the largest or smallest value,For example,let $x(t)=at^2+bt+c$,when we differential it with t and set the differential formula be equal to zero,and we ...
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### Find global minima of vector valued quadratic equation [closed]

I have the following equation, with 5 $\mathbb{R}^3$ vectors A, B, C, D, and P, and a scalar, $t$; $-At^3 + 3Bt^3 + Ct^3 - 3Dt^3 + 3At^2 - 6Bt^2 + 3Dt^2 - 3At + 3Bt + A - P$ I'm trying to find the ...
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### The set of values of $a$ for which the function does not posses critical points

The set of all values of '$a$' for which the function, $f(x)=(a^2-3a+2)(\cos^2{x/4} - \sin^2{x/4}) + (a-1)x + sin1$ does not posses critical points is: I first differented it to find $f'(x)$, then ...
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### Model the following scenario as a multivariable equation with a set of constraints and find the maxima

Greg wants to maximize his credit card rewards and has narrowed his options down to 3 credit cards. However, Greg only wants to apply for 2 new credit cards to avoid hurting his credit score. Based on ...
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### Question on Local Maxima and Local Minima

Find the set of all the possible values of $a$ for which the function $f(x) = 5 + (a-2)x + (a-1)x^2 - x^3$ has a local minimum value at some $x < 1$ and local maximum value at some $x > 1$ The ...
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### Find range of the function $f(x)=\frac{\left\{x\right\}}{1+(\lfloor x\rfloor)^2}$

Find range of the function $f : \mathbb{R} \to \mathbb{R}$ $$f(x)=\frac{\left\{x\right\}}{1+(\lfloor x\rfloor)^2}$$ My try: Obviously range contains zero, since for integers $\left\{x\right\}=0$ ...
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### Local minimum of an analytic function

This is a follow-up to a previous question of mine. I know that any local minimum $x_0$ of a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. ...
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### Minima of $f(x)$

The miniumum value of $\left(1+ \dfrac{1}{\sin^n \alpha}\right)\left(1+ \dfrac{1}{\cos^n\alpha}\right)$ is? Attempt: I expanded the brackets and then differentiated and set the derivative equal to ...
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### Inflection point could not be a local extreme point? [duplicate]

Why is the statement "An inflection point can not be a local extreme point?" wrong? Isn't a local extreme point either a max or a min only? What is wrong here?
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