4.7 Applied optimization problems  (Page 6/8)

When you find the maximum for an optimization problem, why do you need to check the sign of the derivative around the critical points?

Why do you need to check the endpoints for optimization problems?

True or False . For every continuous nonlinear function, you can find the value $x$ that maximizes the function.

True or False . For every continuous nonconstant function on a closed, finite domain, there exists at least one $x$ that minimizes or maximizes the function.

To carry a suitcase on an airplane, the length $+\text{width}+$ height of the box must be less than or equal to $62\text{in}.$ Assuming the height is fixed, show that the maximum volume is $V=h{\left(31-\left(\frac{1}{2}\right)h\right)}^2.$ What height allows you to have the largest volume?

You are constructing a cardboard box with the dimensions $\text{2 m by 4 m}.$ You then cut equal-size squares from each corner so you may fold the edges. What are the dimensions of the box with the largest volume?

Find the positive integer that minimizes the sum of the number and its reciprocal.

Find two positive integers such that their sum is $10,$ and minimize and maximize the sum of their squares.

You have $400\text{ft}$ of fencing to construct a rectangular pen for cattle. What are the dimensions of the pen that maximize the area?

You have $800\text{ft}$ of fencing to make a pen for hogs. If you have a river on one side of your property, what is the dimension of the rectangular pen that maximizes the area?

You need to construct a fence around an area of $1600\text{ft}.$ What are the dimensions of the rectangular pen to minimize the amount of material needed?

Two poles are connected by a wire that is also connected to the ground. The first pole is $20\text{ft}$ tall and the second pole is $10\text{ft}$ tall. There is a distance of $30\text{ft}$ between the two poles. Where should the wire be anchored to the ground to minimize the amount of wire needed?

[T] You are moving into a new apartment and notice there is a corner where the hallway narrows from $\text{8 ft to 6 ft}.$ What is the length of the longest item that can be carried horizontally around the corner?