Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine the interval on which \(f(x)=-6 x^{2}-19 x+7\) is decreasing.

Short answer

The function is decreasing on \((- \frac{19}{12}, \, \infty)\).

Step-by-step solution

- Find the derivative of the function

To determine where the function is decreasing, calculate the derivative of the given function. The function is given by: \[ f(x) = -6x^2 - 19x + 7 \] Differentiate this function with respect to x: \[ f'(x) = \frac{d}{dx}[-6x^2 - 19x + 7] = -12x - 19 \]

- Set the derivative less than zero

A function is decreasing where its derivative is less than zero. So, set the derivative less than zero: \[ f'(x) = -12x - 19 < 0 \]

- Solve the inequality

Solve the inequality for x: \[-12x - 19 < 0 \]Add 19 to both sides: \[-12x < 19 \]Divide both sides by -12 (and remember to reverse the inequality sign when dividing by a negative number): \[x > -\frac{19}{12} \]

- State the interval

The function is decreasing on the interval where x is greater than \(-\frac{19}{12}\). In interval notation, this is represented as: \((- \frac{19}{12}, \, \infty)\)

Key concepts

derivative

The derivative of a function is a measure of how the function's output changes as its input changes. It is often represented as \( f'(x) \) for a function \( f(x) \). The derivative tells us the slope of the tangent line to the function at any given point. In simpler terms, it shows us the rate at which the function is changing.
To find the derivative of a polynomial like \( f(x) = -6x^2 - 19x + 7 \), you use basic derivative rules:

• The power rule: For any term \( ax^n \), the derivative is \( anx^{n-1} \).
• Summation and difference rules: Derivatives of sums and differences are just the sum or difference of the derivatives.

Applying these rules, we get:
\[ f'(x) = \frac{d}{dx}[-6x^2 - 19x + 7] = -12x - 19 \]
It’s crucial to understand derivatives because they help us analyze the behavior of functions, such as finding intervals where the function increases or decreases.

inequality

An inequality is a mathematical statement that shows the relationship between two expressions that are not necessarily equal. When working with derivatives, inequalities are used to identify where a function increases or decreases.
In this exercise, we determined that the function \( f(x) \) is decreasing where its derivative is less than zero:
\[ f'(x) = -12x - 19 < 0 \]
Solving inequalities involves several steps:

• Isolate the variable on one side. For example:
  \[ -12x - 19 < 0 \]
• Move constants to the other side:
  \[ -12x < 19 \]
• Divide by the coefficient of the variable, remembering to reverse the inequality when dividing by a negative number:
  \[ x > -\frac{19}{12} \]

Understanding inequalities allows you to solve for ranges of values that satisfy the inequality. This is essential when determining intervals of decrease or increase for a function.

interval notation

Interval notation is a way of describing a set of numbers along a number line. This notation is especially useful when describing where a function is increasing or decreasing.
In this context, we use interval notation to specify where the function \( f(x) \) is decreasing. After solving the inequality \[ x > -\frac{19}{12} \], the interval of decrease is all numbers greater than \[ -\frac{19}{12} \]:
\[ (- \frac{19}{12}, \, \infty) \]

• Parentheses \( () \) indicate that the endpoint is not included.
• Infinity symbols \( \infty \) are always enclosed in parentheses since infinity is not a specific number that can be included.

By understanding interval notation, you can clearly communicate the ranges of variable values that satisfy particular conditions, such as where a function decreases or increases.