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36 problems match “optimization”. Clear

Minimum of a trigonometric sum ●●●○○ 50 ptsFor real numbers $x$ for which all six functions are defined, find the minimum value of $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x\right|$ . Round your answer to two decimal places.
Smallest bound on derivative at zero ●●●○○ 50 ptsFind the smallest real number $A$ such that for every quadratic polynomial $f(x)$ satisfying $|f(x)| \leqslant 1$ for all $0 \leqslant x \leqslant 1$ , the inequality $f'(0) \leqslant A$ holds.
Optimal bid for an object ●●●○○ 50 ptsThe owner's value $V$ of an object is uniformly distributed on $[0, 100]$ dollars, and its value to you is $1.8V$ . You make one sealed bid of $b$ dollars. If $b$ is more than the owner's value, he sells it to you and you pay $b$ ; otherwise you get nothing. Choose $b$ to maximise your expected profit. What bid $b$ (in dollars) should you make?
Longest girder around a corner ●●●○○ 50 ptsA steel girder of negligible width is moved along a passage 27 feet wide and then around a right-angle corner into a corridor 64 feet wide. Find the length of the longest girder that can be moved around the corner.
Smallest bound on function differences ●●●○○ 50 ptsThe function $f(x)$ is defined on $[0,1]$ and satisfies $f(0)=f(1)$ , and for any $x, y \in [0,1]$ with $x \neq y$ , we have $|f(x)-f(y)|<|x-y|$ . Find the smallest real number $m$ such that for every such function $f$ and any $x, y \in [0,1]$ , we have $|f(x)-f(y)|<m$ . Give your answer as a decimal.
Maximum of tan beta under a constraint ●●●○○ 50 ptsGiven $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$ with $\alpha+\beta \neq \frac{\pi}{2}$ and $\sin \beta=2 \cos (\alpha+\beta) \sin \alpha$ , find the maximum value of $\tan \beta$ . Round to two decimal places.
Largest edge of a box ●●●○○ 50 ptsAn $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are positive integers with $m\le n\le p$ . Find the largest possible value of $p$ .
Maximum pencils with cashback ●●●○○ 50 ptsVasya's mother gave him exactly enough money to buy 30 pencils. All pencils cost the same. A store promotion works as follows: for a receipt for a set of 20 pencils, the factory returns 25% of the set's cost, and for a receipt for a set of 5 pencils, it returns 10% of the set's cost. The refunds can be spent on further purchases. What is the maximum number of pencils Vasya can buy?
Cuboid of minimal surface area ●●●○○ 50 ptsA juice box is a rectangular box (cuboid) with volume $200\ \text{cm}^3$ ( $200$ ml). Find the length $l$ , width $w$ and height $h$ (in cm) of the box with the least possible surface area. Round each answer to five decimal places.
Minimal surface area for fixed volume ●●●○○ 50 ptsA regular right pentagonal prism has a volume of $278.3105$ cubic cm. Let $x$ be the side length of the regular pentagonal bases and $y$ the length (height) of the prism, in cm. Find the values of $x$ and $y$ that minimize the total surface area. Use the area of a regular pentagon $\approx 1.720477\,x^2$ . Round each value to four decimal places.