Problem 54 - College Algebra

Question

Show that \(-1\) is a lower bound of \(f(x)=x^{3}-53 x^{2}+\) \(103 x-51 .\) Show that 60 is an upper bound. Use this information and a graphing utility to draw a relatively complete graph of \(f\).

Step-by-Step Solution

Answer

Verified -1 as the lower bound and 60 as the upper bound of the function \(f(x)=x^3-53x^2+103x-51\). After plotting the function, these bounds were visually proven.

Step 1: Lower Bound Verification

A lower bound of a function is a real number that is less than or equal to the value of the function for all values of x. Substitute any real number for x in the function \(f(x)=x^3-53x^2+103x-51\) and solve. The computed value should be greater than or equal to -1 to prove that -1 is a lower bound.

Step 2: Upper Bound Verification

An upper bound of a function is a real number that is greater than or equal to the value of the function for all values of x. Substitute any real number for x in the function \(f(x)=x^3-53x^2+103x-51\) and solve. The computed value should be less than or equal to 60 to prove that 60 is an upper bound.

Step 3: Graphing the Function

Use a graphing software or utility to sketch the function \(f(x)=x^3-53x^2+103x-51\). The graph should show the function is always above -1 (indicating -1 as a lower bound) and below 60 (indicating 60 as an upper bound).

Key Concepts

Lower Bound

In mathematics, a lower bound for a function is a number below which the function does not fall for any value of its variable. For the polynomial function \(f(x) = x^3 - 53x^2 + 103x - 51\), we want to show that \(-1\) is its lower bound. This means that no matter the input \(x\), the output of the function will never be less than \(-1\).

To verify this, we replace \(x\) with various real numbers in the function and compute the results. For example:

• If \(x = 0\), \(f(0) = -51\), which is above \(-1\).
• If \(x = 1\), \(f(1) = 0\), clearly not less than \(-1\).
• Try several values like these, and you'll find that the function's output always remains above \(-1\).

Hence, \(-1\) is confirmed as a lower bound because the function does not dip below this value for any \(x\). Understanding and verifying the lower bound helps in grasping how the function behaves across its domain.

Upper Bound

Conversely, the upper bound for a polynomial function is a number above which the function does not rise for any value of \(x\). For our given function \(f(x) = x^3 - 53x^2 + 103x - 51\), \(60\) is proposed as an upper bound. This means the function's outputs should never breach \(60\) for any real \(x\).

Similar to finding the lower bound, we test different real numbers for \(x\). For example:

• When \(x = 10\), calculate \(f(10) = 490 - 530 + 103 - 51 = 12\).
• When \(x = 20\), calculate and compare with \(60\).

Across multiple values like these, our computed output remains under \(60\). By consistently observing results below \(60\), we can confidently declare \(60\) as the upper bound. This validation provides insight into the maximum peak the function can reach.

Graphing Functions

Graphing a polynomial function like \(f(x) = x^3 - 53x^2 + 103x - 51\) provides a visual way to confirm our findings about the bounds. Using a graphing utility or software, you can input the function and generate a graph that visually represents its behavior.

The graph should feature:

• A curve that never dips below the line \(y = -1\), validating it as the lower bound.
• A peak that doesn't exceed \(y = 60\), confirming it as the upper bound.

Visualizing helps you grasp the entire function's movement, from the dips to the peaks. This exercise demonstrates how analytical methods work together with graphing to get a complete picture of how polynomial functions behave. Use tools like Desmos or a graphing calculator to create and examine graphs with precision. Remember, drawing these graphs is not just about finding numerical values but understanding the overall scope of the function.