The terms Local Maxima and Minima are used to describe points on a function where it achieves its highest or lowest value within a specified interval. Essentially, a local maximum is an elevation point that surpasses all adjacent points, whereas a local minimum is a point that is lower than any of its immediate neighbors. Understanding these points is pivotal in comprehending the behavior of a function.
| Topic | Problem | Solution |
|---|---|---|
| None | A rectangular page is to contain 27 square inches… | Let the width of the printed area be \(x\) inches and the height be \(y\) inches. Then the area of … |
| None | Two sides of a triangle are 8 and 7 . Find the si… | Given that two sides of a triangle are 8 and 7, we need to find the size of the angle $\theta$ (in … |
| None | Find the relative maximum value of $f(x, y, z)=x … | Define the function to be optimized as \(f(x, y, z) = xyz^2\) and the constraint function as \(g(x,… |
| None | The number of times a new pop song has been downl… | The function given is \(f(t)=\frac{3,900,000}{1+400 e^{-0.75 t}}\). |
| None | The number of times a new pop song has been downl… | The rate of change of the number of downloads is given by the derivative of the function \(f(t)\). … |
| None | Given the function $g(x)=6 x^{3}+27 x^{2}-180 x$,… | Given the function \(g(x)=6 x^{3}+27 x^{2}-180 x\), find the first derivative, \(g^{\prime}(x)\). |
| None | An open-top rectangular box is being constructed … | Let the width of the front (and back) be denoted as x, the depth as y, and the height as z. The vol… |
| None | Find the relative maximum and minimum values. \[ … | First, we need to find the critical points of the function. The critical points are where the first… |
| None | An open-top cylindrical container is to have a vo… | The volume of a cylinder is given by the formula \(V = \pi r^2 h\) where \(r\) is the radius and \(… |
| None | A rectangular box with a volume of $540 \mathrm{f… | The problem is asking for the dimensions of a box that will minimize the cost of construction given… |
| None | A waste management company is designing a rectang… | The dimensions of the dumpster are twice as long as it is wide, so the length is \(2x\), and the wi… |
| None | A zoo supplier is building a glass-walled terrari… | Let the dimensions of the terrarium be x, y, and z. The volume of the terrarium is given by the pro… |
| None | Part 1 of 5 Points: 0 of 1 increasing or decreasi… | To find the relative extrema, we first need to find the derivative of the function \(q(x)\). |
| None | Find the relative extreme points of the function,… | Given the function \(F(x)=\sqrt[3]{x+5}\), we need to find the relative extreme points of the funct… |
| None | A piece of wire of length 57 is cut, and the resu… | Let's denote the length of the wire used to form the circle as \(x\), and the length of the wire us… |
| None | A rectangular tank with a square base, an open to… | The problem is asking for the dimensions of a rectangular tank with a square base that has the mini… |
| None | A pomegranate is thrown from ground level straigh… | The pomegranate hits the ground when its height is zero, i.e., when \(f(t) = 0\). We can solve the … |
| None | Given the cost function $C(x)=3 x^{3}-4 x^{2}+13 … | First, we need to find the marginal cost, which is the derivative of the cost function. So, we have… |
| None | a. A rectangular pen is built with one side again… | Let the sides perpendicular to the barn be of length \(x\). Notice that there are a total of \(100\… |
| None | Use the graph of the given function to find any r… | Given the function \(f(x)=x^{3}-3 x^{2}+1\), we want to find any relative maxima and minima. |
| None | The number of times a new pop song has been downl… | The rate of change of the number of downloads is given by the derivative of the function \(f(t)\). … |
| None | The function $f(x)=3+2 x+18 x^{-1}$ has one local… | Given the function \(f(x) = 3 + 2x + \frac{18}{x}\) |
| None | Find all the local maxima, local minima, and sadd… | First, we need to find the critical points of the function. The critical points are where the first… |