Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f\). (Use the graphs and transformations of Sections 1.2.)
21. \(f(x) = 1 - \sqrt x \)
The absolute maximum occurs at \(x = 0\). There is no absolute minimum as well as the local extrema.
The given function is \(f(x) = 1 - \sqrt x \).
Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.
Let \(y = f(x)\).
Obtain the values of \(y\) for various values of \(x\) as shown in below table and draw the graph of \(f(x)\) as shown below in Figure 1.
From Figure 1, it is observed that there is no absolute minimum and local extrema as the curve approaches numerically large values.
Notice that the only highest point of the curve occurs at \(x = 0\). Therefore, the absolute maximum value is \(f(0) = 1\).
Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.
(a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. Does it appear that there is a maximum volume? If so, estimate it.
(b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols.
(c) Write an expression for the volume.
(d) Use the given information to write an equation that relates the variables.
(e) Use part (d) to write the volume as a function of one variable.
(f) Finish solving the problem and compare the answer with your estimate in part (a).
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