Question

Increasing/Decreasing Function Test Suppose \(f(x)=x^{3}-7 x^{2}-5 x+35 .\) From calculus, the derivative of \(f\) is given by \(f^{\prime}(x)=3 x^{2}-14 x-5 .\) The function \(f\) is increasing where \(f^{\prime}(x)>0\) and decreasing where \(f^{\prime}(x)<0 .\) Determine where \(f\) is increasing and where \(f\) is decreasing.

Short answer

f(x) is increasing on \((-\infty, -\frac{1}{3})\) and \((5, \infty)\), and decreasing on \((-\frac{1}{3}, 5)\).

Step-by-step solution

Find the critical points

To determine where the function is increasing or decreasing, start by setting the first derivative equal to zero and solving for x: \[ f'(x) = 3x^2 - 14x - 5 = 0 \]Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, a = 3, b = -14, and c = -5.

Apply the quadratic formula

Plug in the values a, b, and c into the quadratic formula. \[ x = \frac{14 \pm \sqrt{196 + 60}}{6} \] \[ x = \frac{14 \pm \sqrt{256}}{6} \] \[ x = \frac{14 \pm 16}{6} \]This results in: \[ x = 5 \quad \text{and} \quad x = -\frac{1}{3} \]

Test intervals around the critical points

Determine the sign of the first derivative in the intervals around the critical points: \[ (-\infty, -\frac{1}{3}), (-\frac{1}{3}, 5), (5, \infty) \]Pick test points within each interval (e.g., -1, 0, and 6 respectively) and substitute them into the first derivative to see the sign.

Evaluate the first derivative at test points

For the interval \((-\infty, -\frac{1}{3})\), choose x = -1: \[ f'(-1) = 3(-1)^2 - 14(-1) - 5 = 3 + 14 - 5 = 12 \]This is positive, so f(x) is increasing on \((-\infty, -\frac{1}{3})\).For the interval \((-\frac{1}{3}, 5)\), choose x = 0: \[ f'(0) = 3(0)^2 - 14(0) - 5 = -5 \]This is negative, so f(x) is decreasing on \((-\frac{1}{3}, 5)\).For the interval \((5, \infty)\), choose x = 6: \[ f'(6) = 3(6)^2 - 14(6) - 5 = 108 - 84 - 5 = 19 \]This is positive, so f(x) is increasing on \((5, \infty)\).

Conclusion

Based on the analysis, f(x) is increasing on the intervals \((-\infty, -\frac{1}{3})\) and \((5, \infty)\), and decreasing on the interval \((-\frac{1}{3}, 5)\).

Key concepts

first derivative test

The first derivative test is a crucial method in calculus to determine where a function is increasing or decreasing. When you have a function, like in our example, the first step is to find its derivative. The derivative, denoted as \(f'(x)\), helps us understand how the function's output changes with its input.

If the derivative \(f'(x)\) is positive (\(f'(x) > 0\)), the function \(f(x)\) is increasing. This means as we move from left to right on the x-axis, the values of \(f(x)\) go up.

Conversely, if the derivative \(f'(x)\) is negative (\(f'(x) < 0\)), the function \(f(x)\) is decreasing, meaning the values of \(f(x)\) go down as x increases.

However, to apply this test effectively, we need to find the critical points—where \(f'(x) = 0\) or is undefined—and then check the behavior of the function around these points.

critical points

To determine the increasing and decreasing intervals for a function, we start by finding its critical points. Critical points occur where the first derivative \(f'(x)\) is zero or undefined. These points are essential because they indicate where the function’s behavior changes.

In our example, the function given is \(f(x) = x^3 - 7x^2 - 5x + 35\). We find the first derivative:

div \(f'(x) = 3x^2 - 14x - 5\). Setting the derivative equal to zero, we get:

\(3x^2 - 14x - 5 = 0\).

Solving this equation using the quadratic formula will give us the critical points. These points separate the different intervals we will need to test using values from within each interval.

quadratic formula

To solve the quadratic equation \(3x^2 - 14x - 5 = 0\) from the previous step, we use the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Here, \(a = 3\), \(b = -14\), and \(c = -5\). Plugging in these values, we get:

\[ x = \frac{14 \pm \sqrt{196 + 60}}{6} = \frac{14 \pm 16}{6} \].

This simplifies to two possible solutions: \(x = 5\) and \(x = -\frac{1}{3}\). Thus, these are our critical points.

With these critical points identified, we divide the number line into three intervals: \((-\frac{1}{3}, 5)\), \((5, \infty)\), and \((-\frac{1}{3}, 5)\), and test the sign of the first derivative in these intervals to determine if the function is increasing or decreasing. By selecting points within each interval and evaluating \(f'(x)\) at these points, we comprehensively understand where the function is increasing or decreasing.