Problem 27 - Essential Calculus

Question

In Exercises \(27-30,\) find any relative extrema of the function. Use a graphing utility to confirm your result. \(f(x)=\sin x \sinh x-\cos x \cosh x, \quad-4 \leq x \leq 4\)

Step-by-Step Solution

Answer

The short answer would be presented after performing the calculations. This should include all critical points and whether they are local maxima or minima. The final confirmation is done using a graphing utility.

Step 1: Finding the Derivative

Firstly, differentiate the given function \(f(x)=\sin(x)\sinh(x) -\cos(x)\cosh(x)\) with respect to \(x\). The derivative of the function is obtained using the product rule and the trigonometric and hyperbolic function derivatives.

Step 2: Setting the Derivative Equal to Zero

Secondly, set the derivative equal to zero in order to find the critical points. Solve the equation to get the possible values of \(x\).

Step 3: Second Derivative Test

Next, find the second derivative of the function. Substitute the critical points into the second derivative. If the second derivative at that point is positive, then the function has a local minimum at that point. If it's negative, then there's a local maximum. If it's zero, this test fails.

Step 4: Confirming with a Graphing Utility

Lastly, confirm your results by plotting the original function within domain \(-4 \leq x \leq 4\). The graph should show the relative extrema at the points you've calculated, indicating that your solutions are correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in understanding periodic phenomena. These include functions such as sine, cosine, and tangent. In this exercise, we focus on the sine and cosine functions, which are periodic with period \(2\pi\). They help model oscillations and waves. To differentiate these functions, remember:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).

When these functions are combined with others, as in our given function \(f(x) = \sin(x)\sinh(x) - \cos(x)\cosh(x)\), use the product rule to differentiate correctly. Understanding the cycle and behavior of trigonometric functions is vital for finding points of interest, like extrema.

Hyperbolic Functions

Hyperbolic functions, such as \(\sinh(x)\) and \(\cosh(x)\), are analogous to trigonometric functions but relate to hyperbolas instead of circles. These functions appear frequently in calculus and higher mathematics. They can be defined as combinations of exponential functions. Specifically:

• \( \sinh(x) = \frac{e^x - e^{-x}}{2} \)
• \( \cosh(x) = \frac{e^x + e^{-x}}{2} \)

The derivatives are:

• \( \sinh'(x) = \cosh(x) \)
• \( \cosh'(x) = \sinh(x) \)

In finding extrema for our function, these derivatives are used in the product rule.

Critical Points

Critical points of a function occur where its derivative equals zero or is undefined. These points are crucial as they are potential locations for relative extrema (local minima or maxima). To find them:

• Differentiate the function.
• Set the derivative equal to zero and solve for \(x\).

In this exercise, once the derivative of the function \(f(x)\) is found, solving \(f'(x) = 0\) gives us the critical points. Not every critical point is an extrema, hence the importance of further testing.

Second Derivative Test

The second derivative test helps to determine the nature of critical points. Once you have the critical points, calculate the second derivative of the original function and evaluate it at those points:

• If \(f''(x) > 0\), the function has a local minimum at that point.
• If \(f''(x) < 0\), there's a local maximum.
• If \(f''(x) = 0\), the test is inconclusive.

In our solution, after finding the critical points of \(f(x)\), we use this test to confirm whether they are maxima, minima, or neither. This step ensures that we accurately classify each critical point and helps in understanding the function's behavior in a given interval.