The process of discovering the absolute maximum and minimum within a specified interval is essentially a search for the function's peak and trough within that range. This is accomplished by examining the function at its critical points and endpoints. By comparing these values, we can determine the absolute maximum and minimum.
| Topic | Problem | Solution |
|---|---|---|
| None | A rectangular garden of area 300 square feet is t… | Let the length of the side that will be surrounded by a fence be denoted as \(x\), and the length o… |
| None | Find the absolute maximum and minimum values of t… | We can write the expression as \(k(x, y) = -(x^{2} - 4x + 4) - (y^{2} - 4y + 4)\) |
| None | Maximize $Q=x^{2} y$, where $x+2 y=50$. | We are given the function \(Q=x^{2} y\) and the constraint \(x+2 y=50\). |
| None | Find the maximum profit and the number of units t… | Define the profit function, \(P(x)\), as the difference between the revenue function, \(R(x)\), and… |
| None | Find the absolute maximum and minimum values of t… | The function given is \(f(x)=1-2 x-4 x^{2}\) and we are asked to find the absolute maximum and mini… |
| None | Find the absolute maximum and minimum values of t… | The function is \(f(x)=x^{2}+\frac{160}{x}\) and we are looking for its absolute maximum and minimu… |
| None | Find the absolute maximum and minimum values of t… | First, we need to find the derivative of the function \(f(x) = x^2 + \frac{160}{x}\). Using the pow… |
| None | A dorm at a college houses 1200 students. One day… | First, we need to find when the flu is spreading the fastest. This is when the rate of change of th… |
| None | A supply company manufactures copy machines. The … | The unit cost $C$ is given by the function $C(x)=0.4 x^{2}-96 x+16,541$. |
| None | Homing pigeons avoid flying over water. Suppose a… | The pigeon's path can be divided into two parts: the flight over water from C to some point S on th… |
| None | A Norman window is a rectangle with a semicircle … | Let the radius of the semicircle be \(r\) and the height of the rectangle be \(h\). The perimeter o… |
| None | Find the dimensions of the open rectangular box o… | Given a sheet of cardboard of dimensions 43 inches by 23 inches, we are to cut out congruent square… |
| None | A lifeguard needs to rope off a rectangular swimm… | Let $x$ be the length of a side of the rectangle perpendicular to the shoreline. The total length o… |
| None | Find the absolute extrema of the function, if the… | The function given is a quadratic function, which is a parabola. The coefficient of \(x^2\) is posi… |
| None | Find the absolute maximum and minimum values of t… | Given the function \(f(x)=\frac{2x}{x^{2}+16}\) over the interval \([-9,9]\). |
| None | Find the absolute maximum and minimum values of t… | The function given is \(f(x) = x^{2} - 8x - 4\) and we are asked to find the absolute maximum value… |
| None | Find the absolute maximum and minimum values of t… | The function given is a quadratic function, and its graph is a parabola opening downwards since the… |
| None | Enhanced Homework Part 6 of 8 where $A_{0}$ is th… | The function given is \(A(t)=\frac{A_{0}}{t^{2}+1}\), where \(A_{0}\) is the initial amount of the … |
| None | A company produces two types of solar panels per … | First, we need to find the profit function. The profit, P(x, y), is given by the revenue, R(x, y), … |
| None | Find the absolute maximum and minimum values of t… | Define the function \(g(x) = 2 \csc(x)\), which is equivalent to \(g(x) = 2/\sin(x)\). |
| None | Find the extremum of $f(x, y)$ subject to the giv… | Define the function and the constraint: \(f(x, y) = 40 - x^{2} - y^{2}\) and \(g(x, y) = x + 6y - 3… |
| None | Find the maximum profit and the number of units t… | Given the revenue function, \(R(x) = 50x - 0.1x^2\), and the cost function, \(C(x) = 5x + 30\), we … |
| None | Find the minimum value of the function $f(x)=x^{2… | The function \(f(x)=x^{2}+6 x+12\) is a quadratic function. The graph of a quadratic function is a … |
| None | Find the maximum profit and the number of units t… | Given the revenue function, \(R(x) = 7x - 3x^{2}\), and the cost function, \(C(x) = x^{3} - 5x^{2} … |
| None | A ball is thrown vertically upward. After $t$ sec… | Given the height function \(h(t)=72 t-16 t^{2}\), we need to find the time at which the ball reache… |
| None | Find the minimum value of the average cost for th… | The cost function is given by \(C(x) = x^3 + 37x + 128\). |
| None | Find the absolute extrema if they exist, as well … | First, we need to find the derivative of the function \(f(x)=\frac{6x}{x^{2}+6}\). |
| None | Find the absolute maximum and minimum values of e… | Find the derivative of the function \(f(x) = 3x^3 - 3x^2 - 3x + 5\). |
| None | Find the points on the ellipse $4 x^{2}+2 y^{2}=1… | We are given the function \(f(x, y) = xy\) and the constraint \(4x^2 + 2y^2 = 1\). We want to find … |
| None | Suppose we seek to optimize the objective functio… | Suppose we seek to optimize the objective function \(f(x, y)\) subject to a constraint of the form … |
| None | An employee's monthly productivity $M$, in number… | An employee's monthly productivity $M$, in number of units produced, is found to be a function of t… |
| None | Find the maximum and minimum values of the functi… | Define the function \(g(\theta)=2 \theta-9 \sin (\theta)\). |
| None | Consider the function $f(x)=2-6 x^{2}, \quad-4 \l… | Consider the function \(f(x)=2-6 x^{2}\), for \(-4 \leq x \leq 2\). |
| None | 10 (b) Use calculus to find the value of $x$ for … | Let the wire be bent into a rectangle with sides of length a and b. The perimeter of the rectangle … |
| None | 7 Seja $f(x)=3 x^{2}-2 x+m$. a) Determine o valor… | \(f(x) = 3x^2 - 2x + m\) |
| None | 30. You have 64 feet of fencing to enclose a rect… | Given 64 feet of fencing, let the length of the plot be L and the width be W. The equation for the … |