# Tagged Questions

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### What to do *rigorously* when the second derivative test is inconclusive?

How do you rigorously check if a point is a local minimum when the second derivative test is inconclusive? Does there exist a way to do this in general for arbitrary smooth (or analytic...) functions? ...
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### Application of a derivative [closed]

A spherical projectile 40 cm in diameter and weighing 32kg is shot directly upward from ground level at 196m/sec. Ignoring air resistance during its flight, what is the max height the ball will ...
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### Calculus Optimization - Finding the minimum cost

In oil pipeline construction, the cost of pipe to go underwater is 60% more than the cost of pipe used in dry-land situations. A pipeline comes to a river that is 1 km wide at point A and must be ...
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### How to minimize values in equations?

If you have the equation $-a \leq \cos(45+d) \leq a$ where $a=\sqrt{\frac{(a+b)^2}{2} + c^2}$ and $(a,b,c)$ is a unit vector. for some $d$, how can you minimize $|d|$ so that the above equation is ...
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### Real estate problem - local maxima

A real estate office manages $50$ apartments in a downtown building. when the rent is $\$900 $per month, all units are occupied. for every$\$25$ increase in rent, one unit becomes vacant. on ...
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### Why is that a risk averse consumer buys the optimum insurance when there is actuarially fair insurance?

I've asked the same question at the Quantitative Finance StackExchange. Consider the following example: "As a risk-averse consumer, you would want to choose a value of x so as to maximize expected ...
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Problem statement Let $u(t) \in L^{2}(0, 1)$ and $J(u) = \int_0^1 tu(t) \int_0^t u(s)dsdt$ Compute first and second Frechet derivatives. Attempted solution $$\begin{split} J(u + h) - J(u) &= ... 1answer 62 views ### What trick to calculate this Frechet derivative? Let u(t) \in L^{2}(0, 1). I need to calculate the first and second Frechet derivatives of$$J(u) = \int_0^1 \left(\int_0^{t^3}u(s)ds\right)^2dt I am completely at a loss here: I know several ...
I do understand that differentiating a linear function (for a maximization) subject to some linear restriction (such as the problem $p=ax+by$ s.t. $cx+dy \leq m$) won't necessarily give me the right ...