Problem 64 - Calculus Early Transcendentals

Question

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$g(x)=\sqrt[3]{x-4}$$

Step-by-Step Solution

Answer

Based on the given function $$g(x) = \sqrt[3]{x-4},$$ we found its second derivative, $$g''(x) = \frac{-2}{9}(x-4)^{-\frac{5}{3}}.$$ We then determined that the function is concave down on the interval $$(-\infty, \infty)$$ and has no inflection points, as the concavity does not change.

Step 1: Take the first derivative of the function

To find the first derivative of the function $$g(x)=\sqrt[3]{x-4},$$ apply the power rule: $$\frac{d}{dx}[x^n]=n \cdot x^{n-1}.$$ First, rewrite the function as: $$g(x) = (x-4)^{\frac{1}{3}}.$$ Taking the first derivative, we get: $$g'(x) = \frac{1}{3}(x-4)^{-\frac{2}{3}}.$$

Step 2: Take the second derivative of the function

Now take the second derivative of the function. Apply the power rule again on the first derivative we just found: $$g''(x)=\frac{d}{dx}[g'(x)] = \frac{d}{dx}\left[\frac{1}{3}(x-4)^{-\frac{2}{3}}\right].$$ Applying the chain rule, we obtain: $$g''(x) = \frac{-2}{9}(x-4)^{-\frac{5}{3}}.$$

Step 3: Find the intervals for concavity and inflection points

To find the intervals of concavity and inflection points, we need to analyze the sign of the second derivative, $$g''(x).$$ Concave up: $$g''(x) > 0$$ Concave down: $$g''(x) < 0$$ Since $$g''(x)=\frac{-2}{9}(x-4)^{-\frac{5}{3}},$$ we can set it to greater than or less than zero to check for concavity. We begin by checking for concave up: $$g''(x) > 0$$ $$\frac{-2}{9}(x-4)^{-\frac{5}{3}} > 0$$ Since the power is odd, this inequality can't hold since the term will always be negative. Next, we check for concave down: $$g''(x) < 0$$ $$\frac{-2}{9}(x-4)^{-\frac{5}{3}} < 0$$ Since the term's power is odd, this inequality will always be true. Thus, the function $$g(x)=\sqrt[3]{x-4}$$ is concave down on the interval $$(-\infty, \infty)$$, and there are no inflection points.

Key Concepts

Concave Up

A function is considered to be concave up on an interval if the graph of the function bends upward like a cup. Imagine filling the graph with water; if the water collects without spilling, the function is concave up on that interval. Mathematically, a function is concave up when its second derivative is positive over that interval.

To visualize this, consider a smiling face; the curvature resembles that of a function that is concave up. If you take any two points on such a curve and connect them with a straight line, the line lies below the curve between these points. This property shines when considering optimization problems, as the local minimum of a function will occur where the function changes from concave down to concave up.

Concave Down

In contrast, a function is concave down if it bends downwards, like an upside-down cup or a frowning face. While filling the graph of a concave down function with water, the water would spill over the sides. The formal condition for a function to be concave down over an interval is that its second derivative is negative throughout that interval.

When you take any two points on the curve and draw a line segment connecting them, the segment will lie above the curve. In terms of applications, this is particularly important for identifying local maxima; when a function's concavity shifts from up to down, a local maximum is typically found.

Second Derivative Test

The second derivative test is a convenient method to determine the concavity of a function and possible inflection points. It involves taking the second derivative of the function and analyzing its sign (positive or negative). A change in sign of the second derivative indicates a potential inflection point, where the function switches concavity.

To carry out the test, you first find the first derivative of the function and then differentiate it once more to get the second derivative. By setting the second derivative equal to zero, you can solve for values which might correspond to inflection points. Then, you check the sign of the second derivative before and after these points to determine the intervals of concavity and to confirm the presence of inflection points.

Power Rule

One of the fundamental tools in calculus is the power rule, used for finding the derivative of a function that is a power of x. The rule is simple: for any function of the form \( f(x) = x^n \), where \( n \) is any real number, the derivative is \( f'(x) = nx^{n-1} \). The power rule is frequently applied not just in its basic form but also in combination with other calculations, such as the product rule and the chain rule.

The power rule makes differentiating polynomials straightforward. For example, if you have a polynomial like \( p(x) = 4x^3 + 5x^2 \), applying the power rule gives us the derivative \( p'(x) = 12x^2 + 10x \), seamlessly finding the rate of change at any point along the polynomial.

Chain Rule

The chain rule is essential when dealing with composite functions. A composite function is a function composed of two or more functions, such as \( f(g(x)) \). Here, \( g(x) \) is a function nested inside another function \( f(x) \). The chain rule states that to differentiate a composite function, you should take the derivative of the outer function and multiply it by the derivative of the inner function.

To demonstrate with an easy example, if we have \( h(x) = (5x+3)^2 \), the outer function is \( f(u) = u^2 \) and the inner function is \( g(x) = 5x+3 \). Applying the chain rule, \( h'(x) = 2(5x+3) \times 5 = 10(5x+3) \). This rule is invaluable because it connects the rate of change between the inner and outer layers of composite functions, allowing us to understand complex relationships in advanced mathematics and applied sciences.