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Chapter 4: Problem 53

Sketch a graph of a function \(f\) that is continuous on \((-\infty, \infty)\) and has the following properties. $$f^{\prime}(x)>0, f^{\prime \prime}(x)>0$$

Short Answer

Expert verified
Based on the solution provided, the graph of the function \(f(x)\) should be continuous, increasing, and concave up throughout the entire domain. Examples of such functions include the exponential function \(f(x) = e^x\) and the quadratic function \(f(x) = x^2\). However, the precise appearance of the graph is not important as long as it illustrates the key properties specified in the problem.

Step by step solution

01

Draw the axes

Begin by drawing a set of Cartesian axes on a piece of paper or using graphing software. Label the horizontal axis as \(x\) and the vertical axis as \(f(x)\).
02

Consider the first derivative

Since \(f^{\prime}(x)>0\), the function is increasing over its entire domain. This means that, as we move from left to right along the graph, the function values will always be increasing.
03

Consider the second derivative

Next, since \(f^{\prime \prime}(x)>0\), the function is concave up over its entire domain. This means that the graph of the function will bend upwards at every point and the slope of the function will always be increasing.
04

Sketch the graph

With the information from Steps 2 and 3, we're now ready to sketch the graph. Begin at a point on the left side of the graph, then steadily increase the height of the curve as you move to the right, maintaining the concave up shape. The final graph should resemble a curve similar to the exponential function \(f(x) = e^x\) or a quadratic function like \(f(x) = x^2\), as these functions satisfy the given conditions. Remember, the purpose of this exercise is to sketch a graph, so your drawing doesn't need to be perfect. As long as it illustrates the key properties of the function, you have successfully completed the exercise.

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