Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=x^{3}-3 x+4 $$

Step-by-Step Solution

Answer

The function \(f(x)=x^3-3x+4\) is increasing in the intervals \((-\infty, -1)\) and \((1, \infty)\), and decreasing in the interval \((-1, 1)\).

Step 1: Find the first derivative of the function

The given function is \(f(x)=x^3-3x+4\). Let's find its first derivative: \[ f'(x)=\frac{d}{dx}(x^3-3x+4) \] Using the power rule, we have: \[ f'(x)=3x^2-3 \]

Step 2: Find the critical points

To find the critical points, we need to find the values of \(x\) when \(\displaystyle f'( x)=-3\). So, we set the equation: \[ 3x^2-3=0 \] Factor out a 3: \[ 3(x^2-1)=0 \] Now, we solve the equation \(x^2-1=0\): \[ x^2=1 \] Taking square roots of both sides, we get: \[ x=\pm1 \] So, the critical points are \(x=-1\) and \(x=1\).

Step 3: Determine the intervals of the function increasing or decreasing

Now, we will analyze the sign of the first derivative in the intervals around the critical points. We have three intervals to analyze: \((-\infty, -1)\), \((-1, 1)\) and \((1, \infty)\). 1. Interval \((-\infty, -1)\): Choose \(x=-2\): \[ f'(-2)=3(-2)^2-3=3(4)-3=9>0 \] Since \(f'(-2)>0\), the function is increasing in the interval \((-\infty, -1)\). 2. Interval \((-1, 1)\): Choose \(x=0\): \[ f'(0)=3(0)^2-3=-3<0 \] Since \(f'(0)<0\), the function is decreasing in the interval \((-1, 1)\). 3. Interval \((1, \infty)\): Choose \(x=2\): \[ f'(2)=3(2)^2-3=3(4)-3=9>0 \] Since \(f'(2)>0\), the function is increasing in the interval \((1, \infty)\). So, the final answer is: - The function is increasing in the intervals \((-\infty, -1)\) and \((1, \infty)\). - The function is decreasing in the interval \((-1, 1)\).

Key Concepts

First Derivative Test

The first derivative test is a critical tool in calculus for determining whether a function is increasing or decreasing at specific points on its graph. By taking the derivative of a function, we obtain another function, called the first derivative, which gives us information about the slope of the tangent line to the graph of the original function at any given point.

For instance, if the first derivative of a function is positive over an interval, this means the original function is increasing on that interval. Conversely, if the first derivative is negative, the function is decreasing. If the derivative is zero, the slope of the tangent line is flat, and we may have found a critical point, which could indicate a local minimum or maximum, or possibly an inflection point.

Applying the Test

After finding critical points by setting the first derivative to zero, you can test the intervals around these points. Simply choose test values in the intervals between the critical points and substitute them into the first derivative. The sign of the result will tell you if the function is increasing (positive result) or decreasing (negative result) in that interval. The first derivative test is particularly effective as it not only helps to find where a function is increasing or decreasing, but also assists in sketching the overall graph of the function.

Critical Points

Critical points are essentially the 'X marks the spot' in the treasure map of a function's graph. They are the values of 'x' where the first derivative of a function is either zero or undefined. At these points, the function may have a local maximum, local minimum, or a point of inflection.

Identifying critical points is a fundamental step in analyzing the behavior of functions. To find them, one typically sets the first derivative equal to zero and solves for 'x'. If the derivative doesn't exist for some 'x' values, those are also considered critical points.

Understanding Their Significance

Critical points provide insight into the potential changes in direction of a function. When you know where these points are, you can further analyze them with tests like the first derivative test to thoroughly understand the function’s behavior around those points. In terms of our function, the critical points at 'x' equals -1 and 1 divide the function into intervals where its behavior changes from increasing to decreasing or vice versa.

Power Rule Differentiation

Differentiation is the process of calculating the derivative of a function, and the power rule is one of the most fundamental rules for performing differentiation on functions that are polynomials. The power rule states that if you have a function of the form 'f(x) = x^n', where 'n' is any real number, the derivative of that function 'f'(x)' is 'nx^(n-1)'.

Applying the power rule is straightforward: For each term in the polynomial, multiply the coefficient by the power and then subtract one from the power of 'x'. This rule greatly simplifies the differentiation process, especially for polynomials.

Example in Practice

For our function 'f(x) = x^3 - 3x + 4', we apply the power rule to each term separately. For 'x^3', we multiply the power 3 by the coefficient of 1 and then decrease the power by one. Thus, the derivative of 'x^3' is '3x^2'. For the term '-3x', we treat 'x' as 'x^1', and after applying the power rule, we get '-3'. The constant '4' has a power of zero and its derivative is zero. This results in the simplified first derivative 'f'(x) = 3x^2 - 3', which we use to find critical points and determine the function’s increasing or decreasing intervals.