Question

Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=f(x-10) \quad g^{\prime}(0) \quad 0 $$

Short answer

The appropriate inequality for \(\(g'(0)\)\) would be \(g'(0)>0\).

Step-by-step solution

Understand the transformation

Here the function \(g(x)\) is defined as \(f(x-10)\). This is a horizontal shift of the function \(f(x)\) to the right by 10 units. Because derivatives measure the rate of change, shifting the function, in this case to the right, will not change the overall behavior of the derivative, just the x-values at which these behaviors occur.

Determine the behavior of \(g'(0)\)

Since \(g(x)\) is a shifted version of \(f(x)\), find the corresponding point on the original function \(f(x)\) that matches up with \(x=0\) on \(g(x)\). Due to the shift of 10 to the right, \(g'(0)\) would align with \(f'(10)\) in the original function, and the derivatives at these points would be equal. Thus, we need to find where \(x=10\) would fall in the original function's categories.

Apply the categories of \(f'(x)\)

Compare \(x=10\) to the categories of \(f'(x)\) given in the exercise - it is greater than 6 and so falls within the range of \((6, \infty)\), where \(f'(x)>0\). So, that means \(f'(10)>0\). Since we've ascertained that \(g'(0)=f'(10)\), therefore, \(g'(0)>0\).\n

Key concepts

Calculus

Calculus is an essential branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Understanding calculus is necessary for studying the rates at which quantities change, which is fundamental in the physical sciences, engineering, economics, and in solving real-life problems. The cornerstone of calculus, the derivative, signifies the rate of change of a function with respect to a variable. It aids in determining the slope of the tangent line at any point on the graph of the function, which in turn provides information on the increasing or decreasing behavior of the function.
Derivatives can also be used to calculate maximum and minimum values of functions, showing their extreme points, which can be critical in optimization problems. In essence, mastery of calculus provides the tools to model and analyze dynamic systems.

Function Transformation

In mathematics, function transformation involves shifting, scaling, reflecting, or rotating the graph of a function. These transformations can affect the input or output of the function. For example, if a function, denoted as f(x), undergoes a horizontal shift to the right by k units, the new function, typically denoted as g(x), would be expressed as g(x) = f(x-k).

Horizontal Shifts and Their Effects on Derivatives

Understanding how these transformations affect the derivative is crucial, as they can change the argument of the function but often do not alter the rate at which the function itself changes. This implies that a horizontal shift, such as the one seen in the exercise, does not alter the signs of the derivative, as it does not affect the slope of the function's graph at any given point. The derivative g'(x) at a particular point relates directly to f'(x+k) at the corresponding point shifted by k units.

Derivative Signs

The sign of a derivative is a direct indication of the behavior of a function. If the derivative of a function f'(x) is positive over an interval, the function is increasing on that interval. Conversely, if the derivative is negative, the function is decreasing. This information is crucial when analyzing the behavior of functions and their graphs.

For the exercise at hand, the positive and negative intervals given for f'(x) convey significant information: f(x) is increasing before x = -4, decreasing between x = -4 and x = 6, and increasing again after x = 6. Such knowledge allows one to understand the slope of the tangent line to the function at any given point and anticipate the behavior of f(x) without even seeing its graph.

Inequalities in Calculus

Inequalities play a significant role in calculus as they often describe the range over which a function increases or decreases. When dealing with derivatives, inequalities are used to depict where a function's rate of change is positive or negative, which can be seen in the original exercise. They can also set the stage for understanding the conditions under which a function meets certain criteria, such as when the function reaches its maximum or minimum values within certain boundaries.

Applying Inequalities to Determine Behavior

In applying this to our problem, observing that g'(0) must be greater than zero tells us that at x = 0, the function g(x) is increasing, given that g'(x) corresponds with f'(x+10). We used inequalities to match the x-value from the transformed function g(x) to the appropriate interval in f(x) and determine the behavior of the derivative at that point, proving that inequalities are not just static conditions but dynamic tools that help describe the nature of functions in calculus.