Problem 62 - Essential Calculus Early Transcendentals

Question

Show that if \(f(x)=x^{4},\) then \(f^{\prime \prime}(0)=0,\) but \((0,0)\) is not an inflection point of the graph of \(f .\)

Step-by-Step Solution

Answer

(0,0) is not an inflection point because there's no concavity change.

Step 1: Differentiate the function

First, we need to find the second derivative of the function to determine \( f''(0) \). Start by differentiating \( f(x) = x^4 \). The first derivative of the function with respect to \( x \) is \( f'(x) = 4x^3 \).

Step 2: Find the second derivative

Now, differentiate the first derivative \( f'(x) = 4x^3 \) to find the second derivative \( f''(x) \). The second derivative is \( f''(x) = 12x^2 \).

Step 3: Evaluate the second derivative at x=0

Substitute \( x = 0 \) into the second derivative \( f''(x) = 12x^2 \). Compute \( f''(0) = 12(0)^2 = 0 \). This shows that the second derivative at \( x = 0 \) is zero.

Step 4: Determine if (0,0) is an inflection point

To confirm if (0,0) is an inflection point, the concavity of the function must change at \( x = 0 \). We examine the second derivative \( f''(x) = 12x^2 \). For \( x > 0 \) and \( x < 0 \), \( f''(x) \) remains positive, as squares are always non-negative. There is no sign change in \( f''(x) \), indicating there is no change in concavity. Therefore, (0,0) is not an inflection point.

Key Concepts

Second Derivative

When analyzing functions, calculating the second derivative is crucial. It provides deeper insights beyond what the first derivative can offer. In terms of formulas, if you have a given function, say \(f(x) = x^4\), begin by differentiating it to get the first derivative, \(f'(x) = 4x^3\). The second derivative is then found by differentiating the first derivative. So, for \(f'(x) = 4x^3\), we obtain \(f''(x) = 12x^2\). This second derivative can help assess the function's curvature, which is integral to understanding the shape of the graph.Here's a key point:- The second derivative tells us how the rate of change (i.e., the slope) itself is changing.When you evaluate \(f''(0) = 12 \times 0^2 = 0\), it suggests the slope isn't changing at \(x = 0\). However, this isn't sufficient on its own to conclude about concavity changes or inflection points.

Concavity

Concavity is a concept used to determine the direction a function curves. It is closely linked with the second derivative. When the second derivative is positive, it indicates the function is concave up, similar to a smile. When negative, it means the function is concave down, like a frown.In this case of the function \(f(x) = x^4\), the second derivative is \(f''(x) = 12x^2\). Notice that \(f''(x)\) is always non-negative, as squares yield positive values or zero. This implies that:- For \(x eq 0\), \(f''(x) > 0\), which means the graph is always concave up.Given that there's no point where \(f''(x) < 0\), the concavity does not change from positive to negative or vice versa. This continuous positive concavity indicates that there's no change signs in concavity around \(x = 0\), confirming that \((0,0)\) does not serve as an inflection point.

Differentiation

Differentiation is the process of finding the derivative of a function, which is a measure of how a function's value changes as its input changes. It is foundational for calculus and helps in understanding rates of change and the behavior of functions.Let's break it down:- **First Derivative**: This gives the slope of the tangent line at any point on the function. For \(f(x) = x^4\), the first derivative \(f'(x) = 4x^3\) tells you how steep or flat the graph is at a particular \(x\).- **Second Derivative**: Differentiating more, the second derivative \(f''(x) = 12x^2\) gives information about the curvature of the function. This helps in understanding how the slope of the tangent line is itself changing, revealing more about the graph's shape.Through differentiation, especially using the second derivative, you can determine significant characteristics such as maxima, minima, and inflection points. However, for \(f(x) = x^4\), even though the second derivative at \(x = 0\) is zero, the lack of sign change in \(f''(x)\) across \(x\) confirms no inflection point at \((0,0)\). Differentiation, therefore, is more than just formulas; it's about interpreting the behaviors and traits of functions.