Problem 66 - Calculus - AP Edition

Question

Use a graphing utility to make rough estimates of the intervals on which \(f^{\prime}(x)>0,\) and then find those intervals exactly by differentiating. $$f(x)=x^{3}-3 x$$

Step-by-Step Solution

Answer

The derivative \( f'(x) \) is positive on intervals \((-\infty, -1)\) and \((1, \infty)\).

Step 1: Differentiate the Function

To determine intervals where the derivative is positive, start by finding the derivative of \( f(x) = x^3 - 3x \). Using the power rule, the derivative is \( f'(x) = 3x^2 - 3 \).

Step 2: Find Critical Points

To find where \( f'(x) = 0 \), set the derivative equal to zero: \( 3x^2 - 3 = 0 \). Simplify to \( x^2 - 1 = 0 \) and solve for \( x \). This gives the solutions \( x = 1 \) and \( x = -1 \). These are the critical points.

Step 3: Test Intervals Around Critical Points

The critical points divide the number line into intervals: \((-\infty, -1)\), \((-1, 1)\), and \((1, \infty)\). Pick a test point in each interval to determine the sign of \( f'(x) \).

Step 4: Evaluate Sign of Derivative on Each Interval

- For \((-\infty, -1)\), test \( x = -2 \): \( f'(-2) = 3(-2)^2 - 3 = 9 > 0 \).- For \((-1, 1)\), test \( x = 0 \): \( f'(0) = 3(0)^2 - 3 = -3 < 0 \).- For \((1, \infty)\), test \( x = 2 \): \( f'(2) = 3(2)^2 - 3 = 9 > 0 \).

Step 5: Determine Intervals Where \( f'(x) > 0 \)

From the tests, \( f'(x) > 0 \) on intervals \((-\infty, -1) \) and \((1, \infty)\).

Key Concepts

Critical Points

Critical points are essential in understanding the behavior of a function. These are the points on the graph where the derivative is either zero or undefined. In simpler terms, they are potential points of local maxima, minima, or points of inflection, where the slope of the tangent to the curve is flat.
To locate critical points for a function like \(f(x) = x^3 - 3x\), you start by taking its derivative. For this particular function, the derivative is \(f'(x) = 3x^2 - 3\).
Then, you find the points where \(f'(x) = 0\), which results in the equation \(3x^2 - 3 = 0\). Solving this equation simplifies to \(x^2 - 1 = 0\), which gives us the critical points \(x = 1\) and \(x = -1\).

• These points are crucial as they divide the graph into different intervals.
• Analyzing these intervals helps determine where the function is increasing or decreasing.

Understanding critical points helps in sketching the general shape of the graph and predicting where changes in direction might occur.

Interval Notation

Interval notation is a shorthand way of writing the range of values that a function or a variable can take. It is especially useful in describing the intervals where a function is increasing, decreasing, positive, or negative.
An interval is represented in the form \((a, b)\), which includes all numbers between \(a\) and \(b\) but not \(a\) and \(b\) themselves. If you want to include the endpoints, you would use brackets: \([a, b]\).
For the function \(f(x) = x^3 - 3x\), after finding the critical points \(x = -1\) and \(x = 1\), you divide the real number line into intervals: \((-fty, -1)\), \((-1, 1)\), and \((1, fty)\).

• Each interval represents a different sign of the derivative, \(f'(x)\).
• By testing a point in each interval, you can determine if \(f'(x) > 0\) or \(f'(x) < 0\) there.

Interval notation makes these comparisons more concise and is an essential tool in calculus for describing solution sets.

Power Rule

The power rule is a quick and efficient way to differentiate functions that are polynomials. It's a fundamental rule in calculus and is very easy to apply. The power rule states that if you have a function \(f(x) = x^n\), then its derivative \(f'(x) = nx^{n-1}\).
This rule simplifies the process when dealing with polynomial terms. For instance, for \(f(x) = x^3 - 3x\):

• The derivative of \(x^3\) is \(3x^{3-1} = 3x^2\).
• The derivative of \(-3x\) is \(-3 \times 1 = -3\), since \(x\) is the same as \(x^1\).

Therefore, \(f'(x) = 3x^2 - 3\).
Using the power rule allows you to quickly identify the rate of change of functions and is pivotal in finding critical points, which, as discussed, are key in determining the behavior of a function over different intervals.