Problem 66 - Calculus of a Single Variable

Question

Transformations of Functions In Exercises \(63-68\) , assume that \(f\) is differentiable for all \(x .\) The signs of \(f^{\prime}\) are as follows. $$ \begin{array}{l}{f^{\prime}(x)>0 \text { on }(-\infty,-4)} \\\ {f^{\prime}(x)<0 \text { on }(-4,6)} \\ {f^{\prime}(x)>0 \text { on }(6, \infty)}\end{array} $$ Supply the appropriate inequality sign for the indicated value of \(c .\) $$ g(x)=-f(x) \quad g^{\prime}(0) \quad 0 $$

Step-by-Step Solution

Answer

Given information implies that \( g'(0) > 0 \).

Step 1: Determine the Interval for x=0

The interval in which x=0 falls in according to function f(x) is (-4,6). The derivative of f(x) on this interval is negative, based on the given information.

Step 2: Apply Transformation to Function f

g(x) is defined as -f(x). Since taking the negative of a function inverts the sign, the derivative on the same interval (-4,6) for function g(x) is positive.

Step 3: Find Sign of Derivative at Given Point

According to the transformation, g'(0) will have the opposite sign of f'(0). Since f'(0) is negative (because 0 falls in the negative interval of f), the derivative g'(0) is positive. Therefore, g'(0) > 0.

Key Concepts

Differentiable Functions

Understanding differentiable functions is crucial for delving into calculus and analyzing the behavior of various mathematical models. Simply put, a differentiable function is one that has a derivative at each point in its domain. This means that at any given point on the curve of the function, we can find a tangent line that just 'touches' the curve without cutting through it.

Formally, if you have a function f(x), it is said to be differentiable at a point a if the limit \( \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \) exists. This limit is what we call the derivative of f at a, denoted as \( f'(a) \). It’s important for a student to understand that differentiability implies continuity; in other words, if a function is differentiable at a point, then it must also be continuous at that point. However, the converse is not always true; a function can be continuous but not differentiable at certain points, like at sharp corners or cusps of a graph.

Signs of Derivatives

The sign of the derivative of a function provides significant information about the function's behavior on a certain interval. When the derivative, denoted as \( f'(x) \), is positive (greater than zero) on a particular interval, it indicates that the function is increasing on that interval. On the flip side, if the derivative is negative (less than zero), the function is decreasing on that interval.

As in the original exercise, the function \( f \) is increasing on the intervals where \( f'(x) > 0 \) and decreasing where \( f'(x) < 0 \). This concept assists students in visualizing the graph of the function and understanding its ascents and descents. Moreover, it is essential to remember that the derivative's sign flips when the function is multiplied by a negative, as seen with function \( g(x) = -f(x) \). Hence, knowing how transformations like negation can affect the derivative is key in predicting the behavior of transformed functions.

Function Intervals

A function interval refers to a subsection of the domain of the function where certain properties remain consistent. For instance, intervals can be described where the function is increasing, decreasing, or where the output of the function remains positive or negative. Interval notation, such as (a, b), [a, b), or (-\infty, c], effectively communicates these subsets, with parentheses indicating that the endpoints are not included, and brackets signifying endpoint inclusion.

In calculus, when we talk about function intervals in the context of derivatives, we refer to where the function is increasing (\(f'(x) > 0\)) or decreasing (\(f'(x) < 0\)). Students should learn to determine these intervals by looking at the signs of the derivatives, which act as clues to the behavior of the original function. Identifying these intervals is a fundamental step in sketching the graph of the function and in solving optimization problems where you need to find local maxima and minima.