Question

Explain why $f^{\prime }(x)$ could be positive or negative at a point where $f(x)>0$.

Short answer

Explain. Answer: Yes, the derivative of a function can be either positive or negative at a point where the function itself is greater than 0. This is because the derivative, $f^{\prime }(x)$, represents the rate of change of the function, not the function value itself. If the derivative is positive at a point, it means the function is increasing at that point. If the derivative is negative, it means the function is decreasing. Therefore, it is possible for a function with a positive value to have a positive or negative derivative depending on whether it is increasing or decreasing at that particular point.

Step-by-step solution

Understand the meaning of the derivative

The derivative of a function, $f^{\prime }(x)$, gives us information about the rate of change of the function, or how fast the function is increasing or decreasing at a given point. If the derivative is positive, it means the function is increasing at that point; if the derivative is negative, it means the function is decreasing at that point.

Consider the case when the function is increasing

Suppose at some point $x=a$, the function value $f(a)>0$. Imagine that the function is increasing at $x=a$, which means that the slope of the tangent line to the graph at this point is positive. In this case, we can conclude that $f^{\prime }(a)>0$.

Consider the case when the function is decreasing

Now, let's still assume that at $x=a$, the function value $f(a)>0$. However, this time imagine that the function is decreasing at $x=a$, which means that the slope of the tangent line to the graph at this point is negative. In this case, we can conclude that $f^{\prime }(a)<0$.

Explain the result

As demonstrated in steps 2 and 3, it is possible for the derivative of a function, $f^{\prime }(x)$, to be either positive or negative at a point where the function itself, $f(x)$, is greater than 0. The key is to remember that the derivative represents the rate of change of the function and not the function value itself. Just because the function value is positive, it doesn't mean that the function is necessarily increasing or that the derivative must be positive.

Key concepts

Rate of Change

The concept of rate of change is fundamental in calculus and represents how one quantity changes in relation to another. When we look at the derivative of a function, denoted as f'(x), we're investigating the rate at which the function's value is changing at a certain point along its graph. To understand rate of change, consider a car's speedometer: It shows how rapidly the car's position changes over time. Similarly, a function's derivative tells us how quickly the function's value is changing at a particular x-value.

Mathematically, if you have a function f(x), the rate of change at a point x is given by the derivative f'(x). If f'(x) is positive, the function's value is increasing as x increases; if it's negative, the function's value is decreasing. This behavior holds true regardless of whether the original function's value, f(x), is positive or negative at that point.

Tangent Line Slope

The tangent line to a curve at a particular point is a straight line that just 'touches' the curve at that point without cutting through it. It essentially represents the curve's direction at that point. The slope of this tangent line is of particular interest because it provides a numerical value for the steepness or incline of the curve at exactly one point, which is the derivative of the function at that point, f'(x).

In practical terms, imagine tracing your finger along a hill's contour. As you move along a slope, the angle of the hill under your finger represents the slope of the tangent to the hill's profile at each point. In calculus, the slope of the tangent line to a function's graph at any point describes how steep the graph is at that exact location. This slope is crucial for understanding many physical phenomena, such as acceleration (the rate of velocity change) in physics.

Function Increasing or Decreasing

When analyzing functions, it's essential to determine where they are increasing or decreasing. An increasing function is one where, as you move along the graph from left to right, the function's value rises. Conversely, a decreasing function's value falls as you move from left to right. This behavior is intimately related to the function's derivative: a positive derivative, f'(x) > 0, indicates the function is ascending, whereas a negative derivative, f'(x) < 0, signals that it's descending.

Whether a function is increasing or decreasing at particular points can have significant implications in various fields, such as economics, where an increasing function might represent a growing market. It's critical to recall that a function can increase in one interval and decrease in another, and these changes are depicted by the sign of its derivative within those intervals.