Problem 30 - Essential Calculus

Question

Find any relative extrema of the function. Use a graphing utility to confirm your result. \(h(x)=2 \tanh x-x\)

Step-by-Step Solution

Answer

After conducting the derivative tests, the point \(x=0\) seems to be an extrema. However, the second derivative test was inconclusive, so a graphing tool must be used for final confirmation. After graphing \(h(x)=2 \tanh x-x\), it is seen that \(x=0\) is indeed an extremum point.

Step 1: Differentiation

First, the derivative of the function \(h(x)=2 \tanh x-x\) should be found. The derivative of \(h(x)\) is denoted \(h'(x)\) and it is expressed as \(h'(x)=2(1-\tanh^2 x) -1\).

Step 2: Solve for Critical Points

Next, these derivatives should be set equal to zero and solve for x. The solutions are the critical points - possible locations for extrema. Solving \(2(1-\tanh^2 x) -1 = 0\) yields \(x=0\).

Step 3: Second Derivative Test

Obtain the second derivative of the function, \(h''(x)\), and substitute the x-value obtained from Step 2 (\(x=0\)). If \(h''(x)>0\), then \(x=0\) is a local minimum, and if \(h''(x)<0\), then \(x=0\) is a local maximum. The second derivative of \(h(x)\) is \(h''(x)= -4\tanh x (1-\tanh^2 x)\). Substituting \(x=0\) gives \(h''(0) = 0\), hence, the test is inconclusive.

Step 4: Use a Graphing Utility

Use a graphing tool to graph the function \(h(x)\) and observe the extrema.

Key Concepts

Critical Points in Calculus

In calculus, critical points are essential for analyzing and understanding the behavior of functions. These points are where the derivative of a function is either equal to zero or is undefined. To find the critical points of a function, one must simply take the derivative of the function and solve for the values that make this derivative zero. This methodology applies to a wide range of functions, including polynomials, trigonometric functions, and hyperbolic functions like the tanh function.

In the provided exercise, the critical points of the function h(x) were found by differentiating and solving the equation h'(x) = 0. This process led to the discovery of a critical point at x=0, indicating a potential location of an extremum. It's vital for students to note that not every critical point guarantees an extremum; critical points simply tell us where to investigate further with additional tests.

Second Derivative Test

After identifying the critical points, the second derivative test comes into play as a decisive tool for determining whether these points correspond to local maxima, minima, or points of inflection. To perform the second derivative test, one must first compute the second derivative of the function. Then, you assess the sign of the second derivative at each critical point.

If h''(x) is greater than zero at a critical point, then the function has a local minimum there; conversely, if h''(x) is less than zero, it suggests a local maximum. If the second derivative is zero at the critical point, as it happens with the given function h(x) at x=0, the test is inconclusive, and one must resort to other methods, like the first derivative test or graphical analysis, to classify the extremum.

Differentiation of the Tanh Function

The tanh function, which is the hyperbolic tangent function, often appears in calculus problems. Differentiating tanh(x) requires knowledge of hyperbolic functions and their properties. The derivative of tanh(x) with respect to x is sech^2(x), which is also expressed as 1 - tanh^2(x).

In the exercise, the differentiation of the tanh function was pivotal for finding the first and second derivatives of the function h(x). Understanding how to differentiate tanh(x) can enable students to tackle a host of problems involving hyperbolic functions. When dealing with these kinds of functions, it is rewarding to remember their similarities to trigonometric functions, while also respecting their unique characteristics that influence how they are derived and integrated.