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Chapter 5: Problem 71

Find any relative extrema of the function. \(f(x)=\arctan x-\arctan (x-4)\)

Short Answer

Expert verified
Perform these steps to find the relative extrema of the given function.

Step by step solution

01

Differentiate the function

Differentiate \( f(x) = \arctan x - \arctan (x - 4) \) using the chain rule to get the derivative \( f'(x) \). Remember that the derivative of \( \arctan x \) is \( \dfrac{1}{1+x^2} \), and note that the chain rule will be needed for \( \arctan (x - 4) \).
02

Find stationary points

Now, setting the derivative equal to zero will allow us to find which \( x \) values correspond to horizontal tangents, hence possible extrema. Solve \( f'(x) = 0 \) for \( x \).
03

Determine the nature of each stationary point

To distinguish between maximum and minimum, or to detect any inflection points, take the second derivative \( f''(x) \) and check the sign for each \( x \) from Step 2. This second derivative test will determine the natura of each stationary point.

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Most popular questions from this chapter

In Exercises 83鈥�86, evaluate the definite integral using the formulas from Theorem 5.20. $$ \int_{0}^{1} \frac{1}{\sqrt{25 x^{2}+1}} d x $$

Find an equation of the tangent line to the graph of the function at the given point. \(y=\frac{1}{2} \arccos x, \quad\left(-\frac{\sqrt{2}}{2}, \frac{3 \pi}{8}\right)\)

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