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Chapter 10: Problem 22

Show that the function is concave upward wherever it is defined. $$ h(x)=\frac{1}{x^{2}} $$

Short Answer

Expert verified
The second derivative of h(x) is \(h''(x) = \frac{6}{x^4}\), which is positive for \(x \neq 0\). Therefore, the function h(x) = \(\frac{1}{x^2}\) is concave upward wherever it is defined (x 鈮� 0).

Step by step solution

01

Find the first derivative of h(x)

First, let's find the derivative of the function h(x) with respect to x. Given the function: \(h(x) = \frac{1}{x^{2}}\) Rewrite it as: \(h(x) = x^{-2}\) Using the power rule, the derivative is: \(h'(x) = -2x^{-3}\) So, the first derivative is: \(h'(x) = \frac{-2}{x^{3}}\)
02

Find the second derivative of h(x)

Now we will find the second derivative of h(x) by taking the derivative of the function h'(x). Considering the previous result: \(h'(x) = \frac{-2}{x^{3}}\) And rewrite it as: \(h'(x) = -2x^{-3}\) Now, using the power rule again to find the second derivative: \(h''(x) = -2(-3)x^{-4}\) So, the second derivative is: \(h''(x) = \frac{6}{x^4}\)
03

Determine the concavity of the function h(x)

To determine the concavity of the function h(x), analyze the second derivative h''(x). \(h''(x) = \frac{6}{x^4}\) Since the numerator is positive and the denominator is raised to the power of 4, the fraction is always positive wherever the function is defined (x 鈮� 0). As the second derivative is positive for \(x \neq 0\), the function h(x) is concave upward wherever it is defined. Final conclusion: The function h(x) = \(\frac{1}{x^2}\) is concave upward wherever it is defined (x 鈮� 0).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First Derivative
The first derivative of a function represents the rate at which the function's value changes with respect to a change in its input value. It is essential in determining the function's behavior, such as identifying maxima, minima, and points of inflection.

For the function in our exercise, the first derivative of the function is found using the power rule. This involves differentiating power functions of the form f(x) = ax^n, where a is a constant, n is a real number, and x is a variable. The power rule states that the derivative of x^n is nx^(n-1). Applying this to our function, the first derivative of h(x) = x^{-2} is h'(x) = -2x^{-3}, which simplifies to h'(x) = -2/x^{3}.
Second Derivative
The second derivative is the derivative of the first derivative. It tells us about the rate of change of the rate of change of the function鈥攅ssentially, it provides information about the concavity of the function and can help identify points of inflection.

For our function h(x), the second derivative is determined by applying the power rule once again to the first derivative. This gives us h''(x) = 6x^{-4}, which can also be written as h''(x) = 6/x^4. The second derivative is particularly crucial in analyzing the concavity of functions, as we will explore in the next section.
Concave Upward
A function is said to be concave upward if its second derivative is greater than zero over a particular interval. Visually, this means the graph of the function resembles a 鈥楿鈥� shape in that interval.

By examining the second derivative of h(x), we can determine the concavity of the function. Here, the second derivative h''(x) = 6/x^4 is always positive except at x = 0 (where the function is undefined). Therefore, the function h(x) is concave upward wherever it is defined, which means it slopes upwards and the rate of slope increases with x.
Power Rule
The power rule is a quick and simple method for finding derivatives of functions with one term that contains a power of x. If the original function is f(x) = ax^n, then its derivative is f'(x) = anx^{n-1}. It's worth noting that the power rule applies to negative powers as well, making it a versatile tool for calculus.

In the case of our function h(x) = x^{-2}, we used the power rule to find both the first and second derivatives. This involves multiplying the exponent by the coefficient and then decreasing the exponent by one. The power rule is vital for quick differentiation and is frequently used when analyzing the behavior of functions.

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