Problem 45 - University Calculus: Early Transcendentals

Question

Determine all critical points for each function. $$y=x^{2}-6 x+7$$

Step-by-Step Solution

Answer

The critical point is \((3, -2)\).

Step 1: Understand the Problem

Our task is to determine the critical points of the function. Critical points are where the derivative is zero or undefined. For the given function, which is a polynomial, we will set its derivative equal to zero to find these points.

Step 2: Find the Derivative

First, we need to find the derivative of the function. The function given is \(y = x^2 - 6x + 7\). The derivative of a polynomial is found by applying the power rule: \(x^n\) becomes \(nx^{n-1}\). Therefore, the derivative \(y'\) is:\[ y' = 2x - 6 \]

Step 3: Set the Derivative to Zero

To determine critical points, set the derivative equal to zero and solve for \(x\). This will help us identify points where the slope of the tangent line is horizontal (flat).\[ 2x - 6 = 0 \]

Step 4: Solve the Equation

Solve the equation \(2x - 6 = 0\) for \(x\):- Add 6 to both sides: \(2x = 6\)- Divide by 2: \[ x = 3 \]

Step 5: Determine the Critical Point

We found \(x = 3\) as the solution. To find the corresponding \(y\)-value, substitute \(x = 3\) back into the original function:\[ y = (3)^2 - 6(3) + 7 = 9 - 18 + 7 = -2 \] Therefore, the critical point is \((3, -2)\).

Key Concepts

Derivative

The concept of a derivative is foundational in calculus and helps us understand how a function changes. When we talk about a function's derivative, we're asking how the function's output values change in relation to its input values.
If you're thinking of a graph of this function, the derivative at a particular point tells us the slope of the tangent line at that point.
For any function, where the derivative is zero (or undefined), we identify potential critical points. These points are where the function might have a local maximum or minimum, or a point of inflection.

• For example, with the polynomial function given, the derivative is a simple linear equation, which makes it easy to find these critical areas.

Taking the derivative accurately is the first step in solving for a function's behavior at certain points.

Polynomial Function

A polynomial function is an algebraic expression that includes terms built from a variable raised to a power, each multiplied by a coefficient. In simpler terms, imagine a sequence of terms each involving a variable like 'x' raised to different powers and multiplied by some numbers.

Here's what makes polynomials special:

• They are continuous and smooth, meaning you can draw them without lifting your pencil.
• This makes them easier to analyze, especially when we're looking for critical points or intervals of increase or decrease.

In the given function, \(y = x^2 - 6x + 7\), we have a second-degree polynomial, or a quadratic function. It forms a parabolic curve on a graph, opening upwards as indicated by the positive leading coefficient of \(x^2\).
This function presents a perfect practice for finding derivatives and analyzing those critical points.

Power Rule

When finding the derivative of polynomial functions, the power rule is your best friend. It's a straightforward technique that allows you to calculate derivatives without much hassle.

Here's the core idea of the power rule: if you have a term like \(x^n\), where \(n\) is any real number, the derivative of this term is \(nx^{n-1}\). This means we bring the power down as a coefficient and then reduce the power by one.

• For instance, in our function \(y = x^2 - 6x + 7\), applying the power rule, the derivative of \(x^2\) is \(2x^{2-1} = 2x\).
• Similarly, for the term \(-6x\), it becomes simply \(-6\), since the power of \(x\) is originally 1, and the derivative of a constant (like the 7 in our polynomial) is zero.

By understanding and using the power rule effectively, you can quickly find derivatives of polynomial functions, helping you to determine critical points and analyze the function's behavior.