12 votes

derivative of the max function

No, $g(y)$ is not differentiable in general. For example, let $h: \mathbb{R} \to [0, 1]$ be a smooth function with $h(x) = 0$ when $x \leq 0$ and $h(x) = 1$ when $x \geq 1$. (Such a function exists. ...
David Gao's user avatar
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9 votes
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The minimum possible value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^{2}}}$ WITHOUT the use of calculus

First, I'd note that $$\sqrt{58-42|x|}+\sqrt{149-140\sqrt{1-|x|^{2}}} \leq \sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^{2}}}$$ so we can restrict our domain to $[0,1]$. Then as you have done, use a ...
Brian Moehring's user avatar
5 votes
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Given that, $|z_1 + z_2| = 20 \quad \text{and} \quad |z_1^2 + z_2^2| = 16 $ Evaluate difference of minimum and maximum value of$|z_1^3 + z_2^3|$

(The idea similar as in John Bentin's answer .) With $a=z_1 + z_2$ and $b = z_1^2 + z_2^2$ is $$ z_1^3 + z_2^3 = -\frac 12 a (a^2 - 3b) $$ and therefore $$ |z_1^3 + z_2^3| = \frac 12 |a| \cdot |a^2 -...
Martin R's user avatar
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4 votes
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Evaluate $\int_{-\pi}^\pi1+\lim\limits_{n\to\infty}\min(\cos(x),\dots,\cos(nx))dx$ to find the fractal’s area

For almost any choice of $x\in \mathbb{R}$, the set $$ \{\cos(nx) : n\in \mathbb{N} \}$$ is dense $[-1,1]$. Most books on ergodic theory likely have a proof of this fact, and it looks like there is ...
Xander Henderson's user avatar
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3 votes

Given that, $|z_1 + z_2| = 20 \quad \text{and} \quad |z_1^2 + z_2^2| = 16 $ Evaluate difference of minimum and maximum value of$|z_1^3 + z_2^3|$

$$|z_1+z_2|=20$$ $$|z_1^2+z_2^2|+2|z_1z_2|\ge|z_1^2+z_2^2+2z_1z_2|\ge2|z_1z_2|-|z_1^2+z_2^2|$$ $$208\ge|z_1z_2|\ge192$$ Now, $$|(z_1+z_2)(z_1^2+z_2^2)|+|z_1z_2||z_1+z_2|\ge |(z_1+z_2)(z_1^2+z_2^2)-...
Skdmg's user avatar
  • 702
3 votes
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Conjecture: The line joining the incenter and the circumcenter always subtends an obtuse angle at centroid

Denote the incenter by $I$, circumcenter by $O$, the centroid by $G$. Assume that the triangle is not equilateral, so that the points $I, G, O$ are distinct. In order to prove that $\angle IGO>90^\...
timon92's user avatar
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3 votes

Maximizing $6\left(\cos x-x\right)\left(x+\sqrt{x^2+\sin^{2}{x}}\right)$

The maximum is $3$. Proof. We have $$3 - 6(\cos x - x)x = 6x^2 - 6x\cos x + 3 > 6x^2 - 6|x| + 3/2 = \frac32(2|x| - 1)^2 \ge 0, \tag{1}$$ and \begin{align*} &[3 - 6(\cos x - x) x]^2 - \left[6(\...
River Li's user avatar
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2 votes

Comment on points of maxima and minima for $ f(x) = \frac{\sin(\pi x)}{x^2} $

The derivative is $$f'(x)=\frac{\pi x \cos (\pi x)-2 \sin (\pi x)}{x^3}$$ Because of the leading $x$, the solution will be closer and closer to $ \left(n+\frac{1}{2}\right)$. Let $x=\left(n+\frac{1}...
Claude Leibovici's user avatar
2 votes
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What is the maximum and minimum value of the following function?

Rewrite, $$f(x)=\frac18\left(1+\cos(2x)-8\sin(2x)\right)$$ Note that $a\sin x +b\cos x$ has maximum $\sqrt{a^2+b^2}$ and minimum $-\sqrt{a^2+b^2}$. Thus, $$\max f = \frac{1+\sqrt{65}}8$$ $$\min f = \...
Sahaj's user avatar
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2 votes
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Classifying critical points when hessian has det0

$(0,0)$ is a stationary point for which the Hessian is $\begin{pmatrix} 2 & 0 \\ 0 & 0 \end{pmatrix}$, with eigenvalues $2$ and $0$. Now note that $$f(x,0) = x^2(1-x^2).$$ So increasing $x$ ...
kipf's user avatar
  • 2,357
2 votes

Extrema of derivate are where tangent crosses the curve.

By $T$, the author is referring to the linearization of $f$ about $x=x_0$: $$ T(x) = f(x_0) + f'(x_0)(x-x_0) $$ By the Mean Value Theorem, for all $x \in (a,b)$ there exists a point $\xi$ between $x$ ...
Matthew Leingang's user avatar
2 votes
Accepted

How to Maximize $\sum_{k=1}^{n} \frac{M_{k}}{m_{i} - \sum_{p=1}^{k} M_{p}}$

I don't know about the physics, but are you sure you want to find the maximum instead of the minimum? The maximum is easy: $$\sum_{k=1}^n \frac{M_k}{m_i - \sum_{p=1}^k M_p}\le \sum_{k=1}^n\frac{M_k}{...
Just a user's user avatar
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2 votes

Minimum or maximum value of a function in square roots

You can use the Power-Mean Inequality. Let $a_1=a_2=...=a_9=\displaystyle{\frac{x-2}{9}}$ and $a_{10}=4-x$. Then $$\displaystyle{\bigg(\frac{f(x)}{10}\bigg)^2=\bigg(\sum\limits_{i=1}^{10}\frac{a_i^{\...
grj040803's user avatar
2 votes

Given that, $|z_1 + z_2| = 20 \quad \text{and} \quad |z_1^2 + z_2^2| = 16 $ Evaluate difference of minimum and maximum value of$|z_1^3 + z_2^3|$

From the given conditions, we may write $z_1+z_2:=20\mathrm e^{\mathrm i\theta}$ and $z_1^2+z_2^2:=16\mathrm e^{\mathrm i\phi}$ for some angles $\theta$ and $\phi$. It follows that $$z:=z_1^3+z_2^3=\...
John Bentin's user avatar
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2 votes
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If in $\triangle ABC$, $r=1,a=3$,then find least possible area of $\triangle ABC$

Let the center of the circle $O = (0,0)$, $B = (-b,-r)$, $C = (c,-r)$, and $\beta = \angle CBO$, $\gamma = \angle BCO$​. Let the height of triangle (distance between $A$ and $BC$) be $h$. Minimizing ...
Y.D.X.'s user avatar
  • 364
2 votes

If in $\triangle ABC$, $r=1,a=3$,then find least possible area of $\triangle ABC$

Let put the triangle into the coordinate axes such that $B=(0,0)$ and $C=(3,0)$, then the inscribed circle touches $BC$ at point $D=(a,0)$ (where $a\in[0,3]$ can vary), see the picture. Let $b=\angle ...
van der Wolf's user avatar
  • 2,360
2 votes

If in $\triangle ABC$, $r=1,a=3$,then find least possible area of $\triangle ABC$

HINT.- Since area of $\triangle {ABC}$ is equal to $\dfrac{\overline{BC}\times h}{2}$ we can find the smallest possible $h$ and classic construction of the tangents shows that this occurs when $\...
Piquito's user avatar
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2 votes

derivative of the max function

For $x,y \in [-1,1]$, define $f(x,y) = xy$. Then, $g(y) = \max_x f(x,y) = \max (f(-1,y), f(1,y)) =\max (-y, y) = |y|$ which is not differentiable.
chi's user avatar
  • 2,163
1 vote

If in $\triangle ABC$, $r=1,a=3$,then find least possible area of $\triangle ABC$

Here is one way using Lagrangian. You have found out that $\Delta =s$. Using your notations, the Heron formula implies $(s-3)(s-b)(s-c)-s=0$. We want to minimize $s$. The Lagrangian is $$L(b,c,\lambda)...
toronto hrb's user avatar
1 vote

Given that, $|z_1 + z_2| = 20 \quad \text{and} \quad |z_1^2 + z_2^2| = 16 $ Evaluate difference of minimum and maximum value of$|z_1^3 + z_2^3|$

Let us take as new variables $r_1$, the modulus of $z_1$ $i \omega = \tfrac{z_2}{z_1}$. The two conditions become : $$\begin{cases}r_1 |1+i \omega|&=&20\\r_1^2 |1 - \omega^2|&=&16\...
Jean Marie's user avatar
1 vote

Given that, $|z_1 + z_2| = 20 \quad \text{and} \quad |z_1^2 + z_2^2| = 16 $ Evaluate difference of minimum and maximum value of$|z_1^3 + z_2^3|$

You can go through the algebra: Solving simultaneously: $$(z_1+z_2)\overline{(z_1+z_2)}=400\\(z_1^2+z_2^2)\overline{(z_1^2+z_2^2)}=256$$ We get eight solutions for $(z_1,z_2):$ $$(\mp 10 \mp 6\sqrt{3}...
mjw's user avatar
  • 8,692
1 vote

Maximise $f(x,y)=x^2+y^2$ on contraint that looks like infinity sign

You first define the Lagrangian $L(x,y, \lambda) = x^2+y^2 + \lambda (x^2-y^2 - (x^2+y^2)^2).$ Then, the KKT conditions are given by: $$\begin{cases} \nabla f(x^{\star}, y^{\star}) + \lambda \nabla h(...
rcescon's user avatar
  • 208
1 vote
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Maximum and Minimum of a cubic function

There is a slight mistake in your argument: taking the first derivative, making it vanish, and then checking the second derivative is a recipe to find the local maxima/minima of the function, not the ...
Arnav V Raju's user avatar
1 vote

Minimum or maximum value of a function in square roots

Too long for the comment. Domain of $f(x)$ is $[2,4].$ The equation $\;f'(x)=0,\;$ or $\,\dfrac3{2\sqrt{x-2}}=\dfrac1{2\sqrt{4-x}},\;$ has the single real root $\;x=\dfrac{19}5.\;$ Extremes of $\;f(...
Yuri Negometyanov's user avatar
1 vote

Minimum or maximum value of a function in square roots

I'm going to write two solutions. The domain of $f(x)$ is $x\in [2,4]$. solution 1 : As commented by Alex K, considering the derivative helps. We have $$\begin{align}f'(x)&=\frac{3}{2\sqrt{x-2}}-\...
mathlove's user avatar
  • 141k
1 vote

What is the maximum and minimum value of the following function?

(from one of my comments) “Of course, the values of the tangent function can be “pushed through” with some work to obtain max/min values of the function (evaluate sine and cosine functions at the ...
Dave L. Renfro's user avatar
1 vote

minimum of $-(\cos k_1 + \cos k_2 + \cos k_3)$

Some thoughts. Fact 1. Let $k_1, k_2, k_3 \in \mathbb{R}$ with $k_1 + k_2 + k_3 \in [-\pi, \pi]$. Then $-\cos k_1 - \cos k_2 - \cos k_3 \ge -3\cos \frac{k_1 + k_2 + k_3}{3}$. Fact 2. Let $u, v \in [-\...
River Li's user avatar
  • 37.8k
1 vote

minimum of $-(\cos k_1 + \cos k_2 + \cos k_3)$

I confirm your guess for the case $N=3$. By your hint in the comments, $K$ is a constant in $[-\pi, \pi]$ follows, $k_{1..3}$ are almost free. That is why I disbelieved your minimum if $k_{1..3}=K/N$...
m-stgt's user avatar
  • 309
1 vote
Accepted

Maxima and minima of product

The value of x which you get after solving the first derivative are $\frac{a}{2},\frac{5a}{6}$, for which you're supposed to get minima and maxima. If we take $$y=\sqrt[3]{(x-a)(2x-a)^2}$$ Regarding ...
Gwen's user avatar
  • 1,241
1 vote

Multivariable Calculus - Exercise about Lagrange multipliers

OP has derived these : $$ \begin{align}y &= 2x\lambda \tag{EQ 1} \\ 2y+x &= 2y\lambda \tag{EQ 2} \\ 0 &= 2z\lambda \tag{EQ 3} \\ x^{2}+y^{2}+z^{2}-4&= 0 \tag{EQ 4} \end{align} $$ With ...
Prem's user avatar
  • 9,915

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