Question

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

Short 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-by-step solution

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.

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 \(x\).

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.

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.