Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show that the function is concave upward wherever it is defined. $$ f(x)=\frac{1}{x^{4}} $$

Step-by-Step Solution

Answer

To show that the function \(f(x)=\frac{1}{x^4}\) is concave upward wherever it is defined, we find its second derivative: \(f''(x) = 20x^{-6}\). Since \(f''(x) > 0\) for all defined values of x (i.e., \(x \neq 0\)), the function is concave upward.

Step 1: Calculate the first derivative

: First, we need to find the first derivative of the function \(f(x) = \frac{1}{x^4}\). Use the power rule for differentiation: \[ f'(x) = \frac{d(\frac{1}{x^4})}{dx} = -4x^{-5} \]

Step 2: Calculate the second derivative

: Next, we need to find the second derivative of the function, which is the derivative of the first derivative. Again use the power rule for differentiation: \[ f''(x) = \frac{d(-4x^{-5})}{dx} = 20x^{-6} \]

Step 3: Determine the sign of the second derivative

: Now we want to determine the sign of the second derivative, \(f''(x) = 20x^{-6}\). Notice that, since the power of x is negative, the second derivative is always positive for every x ≠ 0: For \(x > 0\), \(f''(x) = 20x^{-6} > 0\) For \(x < 0\), \(f''(x) = 20x^{-6} > 0\) Since the function \(f(x)\) is not defined for \(x = 0\), we only need to consider nonzero values of x.

Step 4: Conclude that the function is concave upward

: Since the second derivative of the function is positive for all defined values of x (i.e., when \(x\neq0\)), we can conclude that the function \(f(x) = \frac{1}{x^4}\) is concave upward wherever it is defined.

Key Concepts

Power Rule Differentiation

The power rule is a straightforward and widely used method in calculus for finding the derivative of a function that involves a power of x. To apply the power rule of differentiation, for a function of the form \(f(x) = x^n\), where \(n\) is any real number, the derivative \(f'(x)\) is calculated as \(f'(x) = nx^{n-1}\).
For example, in our given function \(f(x) = \frac{1}{x^4}\), it's rewritten using negative exponents as \(f(x) = x^{-4}\). By applying the power rule, we take the exponent \(-4\), multiply it by the coefficient (1 in this case), and reduce the exponent by one:

• Multiply: \(-4 \times 1 = -4\)
• Reduce exponent: \(-4 - 1 = -5\)

Thus, the first derivative becomes \(f'(x) = -4x^{-5}\).
This process makes finding derivatives of polynomial functions quick and efficient.

First and Second Derivatives

Derivatives are one of the fundamental concepts in calculus, describing how a function changes as its input changes.The first derivative of a function, \(f'(x)\), represents the rate of change or slope of the function.
For the function \(f(x) = \frac{1}{x^4}\), we calculated the first derivative as \(f'(x) = -4x^{-5}\).This represents how steeply the function rises or falls at any point along \(x\).
Moving to the second derivative, \(f''(x)\), we derive this from the first derivative. It gives insight about the concavity of the function - essentially how the slope itself is changing.

• If \(f''(x) > 0\), the function is concave upward, like a cup facing upwards.
• If \(f''(x) < 0\), the function is concave downward, or like a frown.

For our function, the second derivative is \(f''(x) = 20x^{-6}\), showing that it is positive for \(x eq 0\), hence concave upward throughout its domain.

Functions and Graphs

Understanding functions and their graphs is crucial for visualizing and interpreting calculus concepts. Functions describe a relationship between variables, and their graphs provide a visual representation of this relationship.
The graph of \(f(x) = \frac{1}{x^4}\) is unique in a few ways:

• It is defined for all \(x eq 0\), avoiding \(x = 0\) as it leads to division by zero.
• The function is symmetric about the y-axis, reflecting the presence of even powers of \(x\).
• Since \(f(x)\) is always positive for \(x eq 0\), the graph lies entirely above the x-axis.

Analyzing the concavity tells us more about the curve's behavior - specifically its shape.A positive second derivative indicates concavity upwards, so here, the curve appears like an opening bowl or cup.
Understanding these graph characteristics helps in predicting how the function behaves as \(x\) increases or decreases.