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Questions tagged [extreme-value-theorem]

This tag is for questions relating to "Extreme Value Theorem". The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions.

Extreme Value Theorem : If a function $f(x)$ is continuous on a closed interval $[ a, b]$, then $f(x)$ has both a maximum and minimum value on $[ a, b]$.

  • Knowing existence and computing extrema of a function will solve problems in many settings. Existence is given by the extreme value theorem.
  • The extreme value theorem is used to prove Rolle's theorem.
  • The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.

For more details see https://en.wikipedia.org/wiki/Extreme_value_theorem

158 questions
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Find the max value of $\sqrt{5x - x^2} + \sqrt{18 + 3x - x^2}$

I have this expression: $C = \sqrt{5x - x^2} + \sqrt{18 + 3x - x^2}$ And I need to find the max value of $C$, can anyone help me? I tried something like this: $$ C^2 = 18 + 8x - 2x^2 + 2\sqrt{5x - x^2}\sqrt{18 + 3x - x^2} \Rightarrow C^2 \leq 18 +…
AquaPI
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If f: [2,5] $\rightarrow [4,13]$ is a continuous function prove that there is $c\in [2,5]$ such that $f(c) = 3c-2$

If $f: [2,5] \rightarrow [4,13]$ is a continuous function prove that there is $c\in [2,5]$ such that $f(c)= 3c-2$. I understand that the extreme value theorem is important in this question and that at most if $c=5$ then $3*5-2=13$, but I am…
Nuemann12
  • 71
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1 answer

Stationary points problem

I already made a first derivation of $f\left ( s,t \right )$. For $\frac{\partial f}{\partial s}=4s^{3}-2s-2t$ and for $\frac{\partial f}{\partial t}=4t^{3}-2s-2t$. I have to find the stationary points. I do know what to do. I tried put t from the…
user714814
1
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2 answers

Extreme values for a vector equation

For a question on physics.stackexchange about Does the Ampère-Maxwell law fail for the field of a uniformly moving point charge? with $$ \vec B(P) = \dfrac{\mu_0 q}{4 \pi} \dfrac{1 - v^2/c^2}{[1 - (v^2/c^2) sin^2 \phi]^{3/2}} \dfrac{\vec v \times…
HolgerFiedler
  • 83
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Find the absolute and relative extrema of $f(x,y,)=8xy+y$, over the region $0≤y≤15-x$, $0≤x≤5$

Firstly, I find the critical points: $f_x=8y=0$; $f_y=8x+1=0$ From here, $y=0$ and $x= -{{1} \over 8}$ and I find the point $P(0, -{{1} \over 8})$ which, does not satisfy the condition under $x$, therefore, there are no critical points in the…
Vile
  • 53
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Is there any simple way to find the extreme points to the equation $f(x)= x^2(\sin(1/x)+\cos(1/x))$when $x \in [-1,0) \cup (0,1]$ and $0$ when $x = 0$

Hey I have found the derivative to this equation: $$2x\cdot\sin\left(\frac{1}{x}\right)+2x\cdot\cos\left(\frac{1}{x}\right)+\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)$$ The things is that i can already see that the endpoint of the…
SliVetle
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Why is this successive bisection proof that proves the boundedness theorem for continuous functions correct?

I am reading Theorem 3.11 from the book "Apostol calculus Vol 1". Let $f$ be continuous on a closed internval $[a,b]$. Then $f$ is bounded on $[a,b]$. That is, there is a number $C\ge 0$ such that $|f(x)|\le C$ for all $x$ in $[a,b]$. The book…
Dachuan Huang
  • 151
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Find a max possible value of an undefined function in an interval when derivative is a constant

The function $f(x)$ is continuous and differentiable in $[0,1]$ if $f'(x)\le 10$ for all $x\in[0,1]$ and $f(0)=0$, What is the maximum possible value of $f(x)$ for $x\in [0,1]$ ? Any help would be greatly appreciated, thanks.
Rita Ryan
  • 3