Problem 80 - Calculus Single Variable

Question

For a function \(f\) that is twice differentiable, define \(C(x)=\) \(a \cdot \operatorname{signum}\left(f^{\prime \prime}(x)\right),\) where \(a\) is a positive constant. Explain how the graph of \(C\) may be used to determine the concavity and the points of inflection of the graph of \(f\). Illustrate by plotting $$ f(x)=6 x^{6}+x^{5}-60 x^{4}-35 x^{3}+120 x^{2}+52 x+160 $$ and \(c(x)=300\) signum \(\left(f^{\prime \prime}(x)\right)\) in the viewing window \([-2.3,2.8] \times[-310,310] .\) (The choice of 300 for \(a\) scales the graph of \(C\) so that we can see it in the same viewing window as \(f .)\)

Step-by-Step Solution

Answer

Use the graph of \( C(x) \) to determine where \( f(x) \) changes concavity, which indicates concavity intervals and inflection points.

Step 1: Understanding the Signum Function

The signum function, often denoted as \( \text{signum}(x) \), returns -1, 0, or 1 depending on whether \( x \) is negative, zero, or positive. In this exercise, \( C(x) = a \cdot \text{signum}(f''(x)) \) gives a value that indicates the sign of the second derivative \( f''(x) \). This helps in determining concavity: if \( f''(x) > 0 \), \( C(x) = a \); if \( f''(x) < 0 \), \( C(x) = -a \); and if \( f''(x) = 0 \), \( C(x) = 0 \).

Step 2: Finding the Second Derivative

First, compute the first derivative, \( f'(x) \), of the given function, \( f(x) = 6x^6 + x^5 - 60x^4 - 35x^3 + 120x^2 + 52x + 160 \). Then, differentiate \( f'(x) \) to find the second derivative \( f''(x) \).

Step 3: Graphing f(x) and C(x)

Using a graphing calculator or software, plot \( f(x) \) and \( C(x) = 300 \cdot \text{signum}(f''(x)) \) in the viewing window \([-2.3, 2.8] \times [-310, 310]\). The graph of \( C(x) \) will appear as horizontal lines above or below the x-axis, indicating where \( f(x) \) is concave up or concave down, respectively.

Step 4: Analyzing Concavity and Points of Inflection

By examining the plot of \( C(x) \), you can identify the intervals of concavity and points of inflection for \( f(x) \). Where \( C(x) = 300 \), \( f(x) \) is concave up, and where \( C(x) = -300 \), \( f(x) \) is concave down. Points where \( C(x) = 0 \) are candidates for points of inflection.

Key Concepts

Concavity

When discussing concavity, we're looking at the shape of the function's graph. Specifically, we determine whether the graph curves upwards or downwards between certain intervals. Concavity is directly related to the second derivative of a function. For a twice differentiable function like the one in our exercise, the second derivative tells us where the function is convex or concave:

• If the second derivative, \( f''(x) \), is positive, the function is concave up (looks like a smile or 'U' shape).
• If \( f''(x) \) is negative, the function is concave down (similar to a frown or an 'n' shape).

Hence, determining the sign of \( f''(x) \) is crucial for understanding the graph's concavity.

Points of Inflection

Points of inflection are where a function changes its concavity. This means the graph shifts from curving upwards to downwards or vice versa. For a point \( x = c \) to be considered a point of inflection, the second derivative must change signs at this point:

• The necessary condition for a candidate to be a point of inflection is that \( f''(x) = 0 \) or \( f''(x) ext{ does not exist} \). However, this alone is not sufficient.
• The sufficiency comes from the sign change on either side of the candidate point. The actual change from positive to negative or negative to positive in \( f''(x) \) confirms a point of inflection.

In this exercise, plotting \( C(x) = 300 \cdot \text{signum}(f''(x)) \) helps visually spot these changes, as the graph crosses the x-axis precisely at the points of inflection.

Second Derivative

The second derivative of a function is derived by taking the derivative of the derivative. It gives us valuable information about the function's concavity and potential points of inflection.For a function \( f(x) \), the second derivative, \( f''(x) \), behaves like this:

• The sign of \( f''(x) \) indicates concavity (positive for concave up and negative for concave down).
• The zero points or points where \( f''(x) \) is undefined are candidates for points of inflection.

In this exercise, calculating the second derivative was crucial for sketching the graph of \( C(x) = 300 \cdot \text{signum}(f''(x)) \). This graph of \( C(x) \) provides a simplified visualization of the changes in concavity and highlights the presence of inflection points by showing where the sign of \( f''(x) \) switches.