Problem 16 - Essential Calculus

Question

Find any critical numbers of the function. $$ g(x)=4 x^{2}\left(3^{x}\right) $$

Step-by-Step Solution

Answer

The critical numbers of the function are \( x = 0 \) and \( x = -2/ln(3) \)

Step 1: Differentiate the function using the product rule and chain rule

We can apply the product rule which says that the derivative of a product of two functions is the derivative of the first times the second function plus the first function times the derivative of the second function. It gives \( g'(x) = 8x(3^x) + 4x^2 * 3^x * ln(3) \).

Step 2: Set the derivative equal to zero

Once you have the derivative, set it equal to zero and solve for x. This gives us \( 8x(3^x) + 4x^2 * 3^x * ln(3) = 0 \).

Step 3: Solve the equation

Factoring out common factors gives: \( 4x^{1}3^{x}(2 + xln(3)) = 0 \). Setting each factor equal to zero gives the critical numbers x = 0 and x = -2/ln(3).

Key Concepts

Product Rule

The product rule is an essential method in calculus used to find the derivative of the product of two functions. Understanding this rule is crucial when dealing with tasks that involve differentiating expressions where two functions are multiplied together.
A quick definition: if you have two functions, say \( f(x) \) and \( h(x) \), and you need to find the derivative of their product \( f(x) \cdot h(x) \), the product rule comes to the rescue. According to this rule:

• Differentiate the first function and multiply it by the second function.
• Then, differentiate the second function and multiply it by the first function.
• Finally, sum those two products together.

Mathematically, this rule is represented as:\[ (fg)'(x) = f'(x)g(x) + f(x)g'(x) \] In our example, we applied the product rule to the function \( g(x)=4 x^{2}(3^{x}) \). The derivative results in two parts: the derivative of \( 4x^2 \) times \( 3^x \) and \( 4x^2 \) times the derivative of \( 3^x \). These parts are then summed to give the final derivative used to find the critical numbers.

Chain Rule

The chain rule is another foundational technique in calculus, especially useful for differentiating composite functions. A composite function is simply a function nested within another function, like \( 3^x \) where the exponent involves the variable \( x \).
The essence of the chain rule is that it allows you to differentiate such functions seamlessly. Here is how it works:

• First, differentiate the outer function, treating the inside function as a constant.
• Then, multiply the result by the derivative of the inside function.

The mathematical expression of the chain rule is:\[ (f(g(x)))' = f'(g(x)) \cdot g'(x) \] In our problem, the chain rule was used to differentiate \( 3^x \). The outer function is the exponential function, and the inner function is \( x \) itself. Differentiating gives \( 3^x \cdot \ln(3) \), which was multiplied by \( 4x^2 \) in the product rule, as these together form part of the critical steps to find the derivative used to solve for critical numbers.

Derivative

The derivative is a fundamental concept in calculus, representing the rate of change of a function with respect to its variable. It's like finding how something changes instantaneously at any given point. Learning to compute derivatives is crucial for solving problems involving rates, slopes, and extreme points.
In this particular problem, we were tasked with finding the derivative of \( g(x)=4 x^{2}(3^{x}) \) so that we could then identify the critical numbers of the function.
Critical numbers are values of \( x \) where the derivative equals zero or is undefined, indicating potential maxima, minima, or points of inflection. After finding the derivative using the product and chain rule, we set it to zero to find critical points:

• By simplifying \( 8x(3^x) + 4x^2 * 3^x * \ln(3) = 0 \).
• We discovered the critical points where the behavior of the function changes occur at \( x = 0 \) and \( x = -2/\ln(3) \).

This process highlights how derivatives are used as tools in understanding the dynamics and behaviors of complex functions.