Problem 37 - Calculus Single Variable

Question

\(P\) In Exercises \(37-42,\) the derivative \(f^{\prime}\) of a function \(f\) is given. Determine and classify all local extrema of \(f\). $$ f^{\prime}(x)=x(x-1) $$

Step-by-Step Solution

Answer

Local maximum at \( x = 0 \), local minimum at \( x = 1 \).

Step 1: Identify where the derivative is zero

To determine critical points, where potential local extrema may occur, find where the derivative \( f'(x) = x(x-1) \) equals zero. Solve the equation \( x(x-1) = 0 \) to find \( x = 0 \) and \( x = 1 \) as critical points.

Step 2: Determine intervals to test the sign of the derivative

Use the critical points to divide the number line into intervals: \((-\infty, 0)\), \((0, 1)\), and \((1, \infty)\). We will test a point in each interval to determine the sign of \( f'(x) \) and the behavior of \( f(x) \) within those intervals.

Step 3: Test the sign of the derivative in each interval

- For the interval \((-\infty, 0)\), choose \( x = -1 \): \( f'(-1) = (-1)((-1)-1) = 2 \). Positive.- For the interval \((0, 1)\), choose \( x = 0.5 \): \( f'(0.5) = 0.5(0.5-1) = -0.25 \). Negative.- For the interval \((1, \infty)\), choose \( x = 2 \): \( f'(2) = 2(2-1) = 2 \). Positive.

Step 4: Use sign changes to classify local extrema

- At \( x = 0 \), \( f'(x) \) changes from positive to negative, indicating a local maximum.- At \( x = 1 \), \( f'(x) \) changes from negative to positive, indicating a local minimum.

Step 5: State the classification of extremas

Given the behavior of \( f'(x) \) around the critical points: \( x = 0 \) is a local maximum and \( x = 1 \) is a local minimum.

Key Concepts

Understanding Derivatives

In calculus, the derivative of a function is a fundamental concept that helps us understand how the function behaves at every point. The derivative provides a way to calculate the slope of the tangent line to the curve of a function at any desired point.

• A derivative tells us how a function changes as its input changes.
• Mathematically, the derivative of a function \( f \) at a point \( x \) is often represented as \( f'(x) \).

In our example, the derivative \( f'(x) = x(x-1) \) helps us determine where the function is increasing or decreasing by looking at where this expression equals zero and changes signs.

Identifying Critical Points

Critical points are locations on a graph where the derivative is either zero or undefined. These points play a key role in finding local maxima and minima.

• To find critical points, set the derivative equal to zero and solve the equation.
• For \( f'(x) = x(x-1) \), solving \( x(x-1)=0 \) gives us critical points at \( x = 0 \) and \( x = 1 \).

Once identified, these critical points can indicate potential locations for local extrema, such as peaks or valleys in the function.

Identifying Local Extrema

Local extrema refer to the highest or lowest points in a particular section of a graph. These are termed 'local maximum' or 'local minimum' depending on whether the function changes from increasing to decreasing or vice versa at these points.

• A local maximum occurs at a point where the function changes from increasing to decreasing.
• A local minimum occurs where the function changes from decreasing to increasing.

In our case, the critical point \( x = 0 \) indicates a local maximum, and \( x = 1 \) indicates a local minimum, as determined by the change of sign in the derivative.

Performing a Sign Test

To fully understand the behavior around critical points, a sign test can be performed. This involves evaluating the sign of the derivative in intervals around the critical points to see how the function behaves in those regions.

• Divide the number line using the critical points to create intervals.
• For example, choose test points in the intervals \((-\infty, 0)\), \((0, 1)\), and \((1, \infty)\).

Evaluating the derivative at these points allows us to determine if the function is increasing or decreasing in those intervals. In this exercise, it was found that the function changes from positive to negative at \( x = 0 \) and from negative to positive at \( x = 1 \).

Analyzing Function Behavior

Understanding the overall behavior of a function is an integral part of calculus. This involves studying how the function behaves as \( x \) approaches various critical points and extends to infinity.

• Positive derivatives indicate the function is increasing.
• Negative derivatives show the function is decreasing.

By analyzing intervals and the function’s derivative, one can piece together a more detailed picture of the function’s graph. For \( f(x) \), because \( f'(x) \) changes from positive to negative at \( x = 0 \), the function reaches a peak, or maximum, there. Similarly, a change from negative to positive at \( x = 1 \) suggests the function meets a valley, or minimum.