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### 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. ...
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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 ...
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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 ...
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### 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.
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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 ...
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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 $\;f(... 1 vote ### Minimum or maximum value of a function in square roots I'm going to write two solutions. The domain of$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}}-\... • 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 ... • 36.9k 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 [-\... • 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... • 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 takey=\sqrt[3]{(x-a)(2x-a)^2}$$Regarding ... • 1,241 1 vote ### Multivariable Calculus - Exercise about Lagrange multipliers OP has derived these :$$ \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 ...
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