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Chapter 3: Problem 46

Consider a function \(f\) such that \(f^{\prime}\) is decreasing. Sketch graphs of \(f\) for (a) \(f^{\prime}<0\) and (b) \(f^{\prime}>0\).

Short Answer

Expert verified
When \(f^{\prime}<0\), the graph of the function \(f\) is a downward concave curve since the function is decreasing and slope (its derivative) is getting lesser in negatives. While when \(f^{\prime}>0\), the graph of \(f\) is a concave down curve that is rising, as the function is increasing but its derivative is getting lesser positives. Both graphs indicate that the derivative of \(f\) is decreasing.

Step by step solution

01

Sketch the graph when \(f^{\prime}

When \(f^{\prime}<0\), it implies that \(f\) is a decreasing function. Also, since \(f^{\prime}\) is decreasing, the slope of the function \(f\) is also decreasing, which suggests a downward concave curve. Thus, the graph of the function will be going down, and it's a concave down curve.
02

Sketch the graph when \(f^{\prime}>0\)

On the other hand, when \(f^{\prime}>0\), it signifies that \(f\) is an increasing function. Given that \(f^{\prime}\) is decreasing, the slope of the function \(f\) is becoming less steep, which indicates a concave down curve as well. Thus, the graph of the function will be rising, but it is also a concave down curve.

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Most popular questions from this chapter

Let \(f\) and \(g\) represent differentiable functions such that \(f^{\prime \prime} \neq 0\) and \(g^{\prime \prime} \neq 0\). Prove that if \(f\) and \(g\) are positive, increasing, and concave upward on the interval \((a, b),\) then \(f g\) is also concave upward on \((a, b)\).

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{x^{2}}{x^{2}-1} $$

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In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ g(x)=\sin \left(\frac{x}{x-2}\right), \quad x>3 $$
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