Question

Find the \(x\) -coordinate of the point(s) of inflection of \(f(x)=\tanh ^{2} x\).

Short answer

Answer: There is no point of inflection for the function \(f(x) = \tanh^2x\).

Step-by-step solution

Find the first derivative of the function

To find the first derivative of \(f(x) = \tanh^2x\), we will apply the chain rule. Let \(u = \tanh x\). Then, \(f(x) = u^2\), and we can find the derivatives of \(u\) and \(f\) with respect to \(x\). We have: \(u'(x) = \frac{d}{dx}(\tanh x) = \text{sech}^2(x)\) Now we can differentiate \(f(x)\) with respect to \(x\) using the chain rule: \(f'(x) = \frac{df(u)}{dx} = 2u \cdot \frac{du}{dx} = 2 \tanh x \cdot \text{sech}^2(x)\)

Find the second derivative of the function

After finding the first derivative, we need to calculate the second derivative to find the point of inflection. The second derivative is the derivative of the first derivative: \(f''(x) = \frac{d}{dx} (f'(x)) = \frac{d}{dx} (2 \tanh x \cdot \text{sech}^2(x))\) We can apply the product rule to find the derivative. Using the product rule: \(f''(x) = \frac{d}{dx}(2 \tanh x) \cdot \text{sech}^2(x) + 2 \tanh x \cdot \frac{d}{dx}(\text{sech}^2(x))\) To find the derivatives of the two terms, we first differentiate the hyperbolic tangent function: \(\frac{d}{dx}(2 \tanh x) = 2 \text{sech}^2(x)\) Next, we differentiate the square of the hyperbolic secant function: \(\frac{d}{dx}(\text{sech}^2(x)) = -2 \text{sech}^2(x) \tanh x\) Now we can plug these derivatives into the second derivative equation: \(f''(x) = 2 \text{sech}^2(x) \cdot \text{sech}^2(x) - 2 \tanh x \cdot (-2 \text{sech}^2(x) \tanh x) = 2 \text{sech}^4(x) + 4 \tanh^2x \text{sech}^2(x)\)

Find the x-coordinate of the inflection point(s)

To find the x-coordinate of the point(s) of inflection, we need to find the value(s) of \(x\) when \(f''(x) = 0\) or \(f''(x)\) is undefined. The second derivative equation is: \(f''(x) = 2 \text{sech}^4(x) + 4 \tanh^2x \text{sech}^2(x)\) Notice that \(\text{sech}^2(x)\) and \(\text{sech}^4(x)\) are always positive for any real \(x\), and thus \(f''(x)\) is always positive. Therefore, the second derivative will never be zero or undefined for any real value of \(x\). Thus, there is no point of inflection for the function \(f(x) = \tanh^2x\).

Key concepts

Inflection Points

Inflection points are fascinating features of a function's graph, marking where the curve changes its concavity. In simple terms, a point of inflection occurs where the graph of the function transitions from being concave up (like a cup) to concave down (like a frown), or vice versa. To find these points, you need to look at the second derivative of a function, which tells us how the slope of the function's graph is changing.
For a point to be an inflection point, the second derivative of the function must be zero or undefined at that point. Additionally, the sign of the second derivative should change as you pass through that point. This is key because it's this change in sign that indicates a change in concavity. Always remember to verify the sign change to confirm that it is indeed a true point of inflection.

Derivative

A derivative, in calculus, is a measure of how a function changes as its input changes. It gives you the rate at which a function's value is changing at any given point. The derivative is fundamental in understanding the behavior of functions and is crucial in finding extremum points (like maxima and minima) as well as inflection points.
The process of finding a derivative is called differentiation. When differentiating, you're essentially finding the slope of the tangent line to the function at any point. The first derivative itself can be differentiated again, leading to the second derivative, which offers insight into the function's concavity and potential inflection points.

• The first derivative tells us about the function's increasing or decreasing behavior.
• The second derivative provides information about the curvature or concavity of the function's graph.

Hyperbolic Functions

Hyperbolic functions are analogs of common trigonometric functions but for the hyperbola. Here, the focus is on the hyperbolic tangent function, \(\tanh x\).
These functions are valuable in various fields such as calculus, engineering, and physics. The hyperbolic tangent, specifically, is defined as:

\[ \tanh(x) = \frac{\sinh(x)}{\cosh(x)} \]

Where \(\sinh(x)\) and \(\cosh(x)\) are hyperbolic sine and cosine, respectively. One of the notable properties of the hyperbolic tangent function is that its values are bound between -1 and 1, much like the regular tangent but with smoother boundaries.
Understanding the behavior of hyperbolic functions helps in calculating their derivatives, which follow specific rules. For example, the derivative of \(\tanh(x)\) is \(\text{sech}^2(x)\), a result that is especially significant when working with their derivatives to find changes in concavity.

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. A composite function is one where one function is applied to the result of another function. The chain rule is invaluable when dealing with functions inside other functions, such as \(f(x) = \tanh^2 x\).
In applying the chain rule, you take the derivative of the outer function while minding the inner function's derivative. For our example, \(f(x) = u^2\) where \(u = \tanh x\). When differentiating, we used:

\[\frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \]

This simplifies the process and ensures that all parts of the function are taken into account.

• The first step is to identify the "inner" and "outer" functions.
• Next, differentiate each one separately.
• Finally, combine these derivatives according to the chain rule formula.

Using the chain rule correctly allows us to navigate through complex functions and understanding them deeply is key to mastering calculus.