Question

In Exercises,Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { If } f^{\prime}(c) \text { and } g^{\prime}(c) \text { are zero and } h(x)=f(x) g(x) \text { , then } h^{\prime}(c)=0 . $$

Short answer

The statement is true. The derivative of function \( h(x) \) at point \( c \) is indeed zero if both \( f'(c) \) and \( g'(c) \) are zero.

Step-by-step solution

Clarifying Mathematical Concepts

Firstly recall the product rule of differentiation which states \( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \). This rule will be essential in solving this problem.

Applying the product rule

Using the product rule on the function \( h(x) = f(x)g(x) \), differentiating with respect to \( x \), yields the formula \( h'(x) = f'(x)g(x)+f(x)g'(x) \). We then plug \( x = c \) into this formula to see if we get zero.

Substituting into Formula

Substituting the \( c \) into this formula we get \( h'(c) = f'(c)g(c) + f(c)g'(c) \) . Now, in the problem, it's mentioned that \( f'(c) \) and \( g'(c) \) are zero. Replacing these zero values into the formula results in \( h'(c) = 0*g(c) + f(c)*0 = 0 \).

Concluding the Solution

So, from the above steps, it's clear that the statement 'If \( f'(c) \) and \( g'(c) \) are zero and \( h(x) = f(x)g(x) \), then \( h'(c) = 0 \)' is true.

Key concepts

Product Rule

The Product Rule is an essential concept when dealing with derivatives of functions that are products of two or more functions. When you multiply two functions and want to find the derivative of the resulting function, the product rule is your go-to tool. It helps you compute the derivative without having to expand the functions separately.
For example, if you have functions \( f(x) \) and \( g(x) \), their product \( h(x) = f(x) \cdot g(x) \) can be differentiated using the product rule as follows:

• Start by differentiating the first function: \( f'(x) \).
• Multiply \( f'(x) \) by the second function \( g(x) \).
• Then differentiate the second function: \( g'(x) \) and multiply it by the first function \( f(x) \).
• Add these results together to find \( h'(x) \): \( h'(x) = f'(x)g(x) + f(x)g'(x) \).

This rule allows you to tackle complex differentiation problems methodically. It is particularly useful when both functions are not easily simplified by themselves.

Derivatives

Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. Think of them as a way to find the slope of a function at any given point. This slope signifies how steep or flat the curve is at a specific location.
In mathematical terms, the derivative of a function \( f(x) \) is represented as \( f'(x) \).
Here's why derivatives are important:

• They tell us the rate of change. For instance, in physics, a derivative could represent velocity, the rate of change of position with time.
• They help in finding the maximum and minimum points of a function, which is crucial in optimization problems.
• Understanding derivatives is critical for analyzing the behavior of complex models and functions.

In our exercise, we use derivatives to analyze the specific values of \( f'(c) \) and \( g'(c) \), determining the nature of \( h(x) = f(x) \cdot g(x) \). Differentiating using derivatives shows the combined effect of the slopes of \( f(x) \) and \( g(x) \), enabling us to conclude about \( h'(c) \).

Function Analysis

Function analysis is the process of understanding what a function represents, its output behaviors, and how its rate of change or slope evolves over a domain. This often involves differentiating the function, identifying key characteristics like intercepts, asymptotes, and extremums (maximum or minimum values).
In our example, the focus is on analyzing the function \( h(x) = f(x)g(x) \) through its derivative \( h'(x) \). We use critical points — where the derivatives are zero — to glean insights into the points where the function could achieve local maxima or minima. These critical points are vital to understanding the behavior of the function.

• By evaluating the function's derivative, we are able to determine where the function is increasing or decreasing.
• We use analyses like these to make predictions about overall function behavior over small and large intervals.
• Particularly at \( x = c \), evaluating \( f'(c), g'(c) \) helps to interpret the importance of zero derivative values at this specific point in the context of the product rule.

Understanding these concepts allows you to dissect complex functions comprehensively, predicting behaviors that are not immediately obvious from just the function equation.