Problem 52 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the inflection point(s), if any, of each function. $$ g(x)=2 x^{3}-3 x^{2}+18 x-8 $$

Step-by-Step Solution

Answer

The inflection point of the function \(g(x)=2x^3-3x^2+18x-8\) is at \(\displaystyle\left(\frac{1}{2}, \frac{1}{2}\right)\).

Step 1: Find the first derivative of g(x)

Differentiate the given function with respect to x: $$ g'(x) = \frac{d}{dx}(2x^3-3x^2+18x-8) $$ Using the power rule for differentiation, we get: $$ g'(x) = 6x^2 - 6x + 18 $$

Step 2: Find the second derivative of g(x)

Differentiate the first derivative with respect to x: $$ g''(x) = \frac{d}{dx}(6x^2 - 6x + 18) $$ Using the power rule for differentiation, we get: $$ g''(x) = 12x - 6 $$

Step 3: Set the second derivative equal to zero and solve for x

In order to find the inflection points, set the second derivative equal to zero and solve for \(x\): $$ 12x - 6 = 0 $$ Solve for \(x\): $$ x = \frac{6}{12} = \frac{1}{2} $$

Step 4: Check for sign change in the second derivative

To verify that this value of \(x\) corresponds to an inflection point, we need to check whether the second derivative changes its sign as we pass through \(x = \frac{1}{2}\). Test the intervals \((-\infty, \frac{1}{2})\) and \((\frac{1}{2}, +\infty)\). For \(x\) in \((-\infty, \frac{1}{2})\), let's test \(x = 0\): $$ g''(0) = 12(0)-6 = -6 < 0 $$ For \(x\) in \((\frac{1}{2}, +\infty)\), let's test \(x = 1\): $$ g''(1) = 12(1)-6 = 6 > 0 $$ Since the second derivative changes its sign as we pass through \(x = \frac{1}{2}\), the function has an inflection point at \(x = \frac{1}{2}\).

Step 5: Find the corresponding y-coordinate of the inflection point

To find the y-coordinate of the inflection point, plug the value of x back into the original function: $$ g(\frac{1}{2}) = 2 (\frac{1}{2})^3 - 3 (\frac{1}{2})^2 + 18 (\frac{1}{2}) - 8 $$ Calculate the value of \(g(\frac{1}{2})\): $$ g(\frac{1}{2}) = 2 (\frac{1}{8}) - 3 (\frac{1}{4}) + 9 - 8 = \frac{1}{2} $$

Step 6: Write the inflection point

The inflection point is at \(\displaystyle\left(\frac{1}{2}, \frac{1}{2}\right)\).

Key Concepts

Second Derivative Test

The second derivative test is a mathematical procedure used to determine the concavity of a function at a particular point and to identify possible inflection points. An inflection point is where the function changes concavity, meaning it goes from being concave up (like a cup) to concave down (like a cap), or vice versa.

To apply the second derivative test in finding an inflection point, you need to follow a few key steps. First, find the second derivative of the function by differentiating the first derivative. Next, set the second derivative equal to zero and solve for the variable, which in this case is x. Once you've found the value of x, you'll need to check whether the second derivative changes sign as it crosses this value. If it does change sign, you've identified an inflection point at that value of x.

In the step-by-step solution we dissected, the second derivative, g''(x) = 12x - 6, changes from negative to positive as we pass through x = 1/2. This confirms an inflection point at x = 1/2. It is crucial to check the sign change because a second derivative equal to zero does not always mean there is an inflection point; the sign change is what confirms it.

Power Rule for Differentiation

The power rule for differentiation is one of the most fundamental rules in calculus and is utilized to find the derivative of a function that is a power of x. The rule states that if you have a function f(x) = ax^n, then its derivative f'(x) with respect to x is f'(x) = anx^(n-1).

When you differentiate an equation like g(x) = 2x^3 - 3x^2 + 18x - 8, you apply the power rule to each term separately. This yields the first derivative g'(x) = 6x^2 - 6x + 18 by bringing down the exponent and subtracting one from it for each x-term.

Applying the rule again to g'(x) gives the second derivative g''(x) = 12x - 6, which is crucial in finding inflection points as discussed earlier. It's important to practice this rule as it simplifies the process of differentiation, making it quicker and more intuitive when dealing with polynomial functions.

Finding y-coordinate

Finding the y-coordinate of a point on a function is an essential step after identifying the x-coordinate, especially if you are determining the location of an inflection point. Once you have the x value, the y-coordinate can be found by substituting this value back into the original function.

In our example, after establishing that there is an inflection point at x = 1/2, we determine the corresponding y-coordinate by calculating g(1/2). It involves plugging x = 1/2 into the original equation g(x) and simplifying. Performing these calculations gives us g(1/2) = 1/2.

The inflection point is, therefore, at the coordinates (1/2, 1/2). Understanding how to find the y-coordinate is crucial as it provides the complete information needed for graphing the point or discussing its properties relative to the graph of the function.