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Chapter 15: Problem 55

Find the absolute maximum and minimum values of the following functions on the given region \(R\). \(f(x, y)=\frac{2 y^{2}-x^{2}}{2+2 x^{2} y^{2}} ; R\) is the closed region bounded by the lines \(y=x, y=2 x,\) and \(y=2\)

Short Answer

Expert verified
The absolute maximum and minimum values of the function \(f(x, y)=\frac{2y^2-x^2}{2+2x^2y^2}\) on the closed region \(R\) bounded by the lines \(y=x, y=2x,\) and \(y=2\) can be found by following these steps: 1. Find the partial derivatives \(f_x\) and \(f_y\), which are \(f_x = \frac{-4xy^2(x^2-2y^2)}{(2+2x^2y^2)^2}\) and \(f_y = \frac{4x^2y(2x^2-y^2)}{(2+2x^2y^2)^2}\). 2. Solve the equations \(f_x=0\) and \(f_y=0\) to find the critical points. 3. Check the boundaries of the region \(R\) by substituting the values of the boundary lines in the function: \(y=x\), \(y=2x\), and \(y=2\). 4. Evaluate the function at the critical points and boundary points. 5. Compare the values to find the absolute maximum and minimum values. By comparing all the values obtained from the critical points and boundaries, the highest value is the absolute maximum, and the lowest value is the absolute minimum.

Step by step solution

01

Find the partial derivatives \(f_x\) and \(f_y\)

Compute the partial derivatives of \(f(x, y)\) with respect to \(x\) and \(y\): \(f_x = \frac{\partial}{\partial x}(\frac{2y^2-x^2}{2+2x^2y^2})\) \(f_y = \frac{\partial}{\partial y}(\frac{2y^2-x^2}{2+2x^2y^2})\) After finding the derivatives, we get: \(f_x = \frac{-4xy^2(x^2-2y^2)}{(2+2x^2y^2)^2}\) \(f_y = \frac{4x^2y(2x^2-y^2)}{(2+2x^2y^2)^2}\)
02

Find the critical points

To find the critical points, we need to solve \(f_x=0\) and \(f_y=0\). That gives us: \(-4xy^2(x^2-2y^2) = 0\) \(4x^2y(2x^2-y^2)= 0\) Solve these equations to get the critical points.
03

Check the boundaries of the region R

The region R is bounded by the lines \(y=x, y=2x\) and \(y=2\). We need to consider these boundaries when looking for the absolute maximum and minimum values of the function. 1. For the boundary line \(y=x\), replace \(y\) with \(x\) in the function and find the resulting one-variable function of \(x\). 2. For the boundary line \(y=2x\), replace \(y\) with \(2x\) in the function and find the resulting one-variable function of \(x\). 3. For the boundary line \(y=2\), replace \(y\) with \(2\) in the function and find the resulting one-variable function of \(x\).
04

Evaluate the function at the critical points and boundary points

Evaluate the function at the critical points obtained in Step 2, and inside the region R for the boundaries found in Step 3. Compute the value of the function for each point.
05

Compare the values to find the absolute maximum and minimum

Once you have all the values for the critical points and boundary points, compare the values to find the absolute maximum and minimum values of the function in the given region R. The highest value will be the absolute maximum, and the lowest value will be the absolute minimum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Understanding partial derivatives is crucial when dealing with functions of several variables, like in multivariable calculus. A partial derivative of a multivariable function is its derivative with respect to one variable, while keeping all other variables constant. Imagine you're taking a snapshot of a function's change by just tweaking one variable 鈥� that's what computing a partial derivative is like.

In the context of the given function f(x, y), we calculate the partial derivatives f_x and f_y to determine the rate at which f changes in the direction of each variable independently. This concept is not just a mathematical curiosity; it's essential in physical sciences and economics to understand how a system evolves with respect to its inputs.
Critical Points
Imagine you're on a hilly terrain and you're trying to find the top or bottom of hills without actually walking the path 鈥� critical points are those tops or bottoms in mathematical terms. They occur where the partial derivatives of a function equal zero because these are the points where the function stops increasing or decreasing momentarily, just like pausing at the peak of a hill to catch your breath.

In multivariable calculus, we find the critical points of a function to identify potential candidates for absolute maximum and minimum values. These aren't guaranteed to be the absolute high and lows, but they're important signposts on the map that lead us towards them. To find these points for f(x, y), we solve the equations f_x=0 and f_y=0. The solutions are the coordinates that give us these noteworthy points.
Boundary of Region
When you're house-hunting, you don't just look inside the house; you walk around it to see the borders of the property. Similarly, in multivariable calculus, you must consider the boundary of the region within which you're operating. The boundary of a region is its enclosing edge, like fences around a field.

In finding the absolute maximum and minimum, we examine the values on the boundaries as they can often contain the extreme values we are searching for, even if they're not evident from the critical points found in the interior of the region. Given a specific boundary, we plug these constraints into the function to evaluate it along these edges. In our example, the lines y=x, y=2x, and y=2 bound the region of interest, so we investigate the function's behavior on these lines.
Multivariable Calculus
Multivariable calculus is the gateway to visualizing and analyzing functions that have more than one input. Imagine holding a 3D map that varies in topology, where the altitude represents the value of a function depending on your position on the horizontal plane 鈥� this is the essence of functions with multiple variables.

It extends the notions of single variable calculus to higher dimensions, and this means generalizing key concepts like derivatives, integrals, and limits to functions of multiple variables, such as f(x, y) in our exercise. This area of mathematics is indispensable for understanding and modeling phenomena where several factors interact at the same time: it's the mathematics of the real, multifaceted world.

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Most popular questions from this chapter

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