Problem 19 - Calculus for Biology and Medicine

Question

In Problems 19-26, find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=x^{2}\)

Step-by-Step Solution

Answer

At \(x=0\), \(f(x) = x^2\) has a local minimum since \(f''(x)\) is positive.

Step 1: Differentiate the function

To find where the derivative is zero, start by differentiating the function. The given function is \(f(x) = x^2\). Differentiate this to get \(f'(x) = 2x\).

Step 2: Set the derivative to zero

Set the derivative found in the previous step equal to zero to find the values of \(x\) where the slope is zero. So, set \(2x = 0\).

Step 3: Solve for \(x\)

Solve the equation \(2x = 0\). Dividing both sides by 2 gives us \(x = 0\). This is the critical point where \(f'(c) = 0\).

Step 4: Determine the nature of the extremum using the second derivative test

To determine if there is a local extremum at \(x = 0\), use the second derivative test. First, find the second derivative \(f''(x)\). Differentiating \(f'(x) = 2x\) yields \(f''(x) = 2\). Since \(f''(x) = 2\) is positive, \(f(x)\) has a local minimum at \(x = 0\).

Key Concepts

Derivative

The concept of a derivative is foundational in calculus as it represents the rate of change or the slope of the tangent line of a function at a given point. In simpler terms, the derivative tells us how a function is changing at any point on the graph. For the function \(f(x) = x^2\), its derivative \(f'(x) = 2x\) was found by the rules of differentiation. Differentiation involves applying formulas to find the first derivative, which is crucial for identifying where the function reaches a high or low point. Understanding how to compute and interpret derivatives helps in analyzing the behavior of functions.

Critical Point

A critical point is where a function's derivative is either zero or undefined. These points are essential because they indicate where a function's graph may have a peak, valley, or a flat region. For \(f(x) = x^2\), the critical point occurs at \(x = 0\) because this is where \(f'(x) = 2x\) equals zero. Critical points mark potential local maxima, minima, or saddle points, which are the subject of further testing to ascertain their nature. Knowing how to find and interpret critical points is vital for graph sketching and optimization problems.

Second Derivative Test

The second derivative test is a handy tool to verify the nature of critical points. This method uses the second derivative of the function to determine concavity: whether a function is curving upwards or downwards. For the quadratic function \(f(x) = x^2\), the second derivative was found to be \(f''(x) = 2\), which is a positive constant. Since the second derivative is positive, it confirms that the graph is concave up at \(x = 0\), identifying a local minimum. Thus, the second derivative test quickly helps to classify the type of extremum present at a critical point.

Local Extremum

A local extremum refers to the highest or lowest points in a local neighborhood of a function. Local extrema are identified as either minimums or maximums. In the example of \(f(x) = x^2\), it was concluded that there is a local minimum at \(x = 0\). The graph of \(x^2\) goes down to this point and then rises again, creating a "valley". Local extrema are essential for understanding the overall shape of a graph, optimizing functions, and providing insights into real-world phenomena modeled by mathematical functions.