Problem 1 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a table similar to that in Example 1 to find all relative extrema of the function. $$ f(x)=-2 x^{2}+4 x+3 $$

Step-by-Step Solution

Answer

The function \(f(x)= -2x^2+4x+3\) has a local maximum at \(x=1\).

Step 1: Find the Derivative

The derivative of the function \(f(x)=-2x^2+4x+3\) can be found using the power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\). Here, the derivative of \(f(x)\) is \(f'(x)=-4x+4\).

Step 2: Solve the Derivative Equal to Zero

Setting the derivative equal to zero will give us the critical numbers, which are potential points where extrema occur. So we solve the equation \(-4x+4=0\), and we get \(x=1\).

Step 3: Apply the First Derivative Test

To determine whether the critical number \(x=1\) corresponds to a local maximum, minimum or neither, we check the sign of the derivative to the left and right of \(x=1\). If \(f'(x)\) changes from positive to negative at \(x=1\), \(x=1\) is a local maximum. If \(f'(x)\) changes from negative to positive at \(x=1\), \(x=1\) is a local minimum. If the sign does not change, \(x=1\) is neither a maximum nor a minimum. In this case, to the left of x=1 (let's say x=0), \(f'(0)=4>0\). To the right of x=1 (let's say x=2), \(f'(2)=-4<0\). The function changes from increasing to decreasing at \(x=1\). Hence, \(x=1\) is a local maximum.

Key Concepts

Derivative of a Function

Understanding the derivative of a function is a vital part in calculus, particularly when finding relative extrema. Simply put, a derivative represents the rate of change or the slope of the curve of a function at a particular point. It's akin to figuring out how steep a hill is when you're standing at a specific point.

For our exercise, we deal with the function f(x)=-2x^2+4x+3. To find its derivative using the power rule—which states that the derivative of x^n is nx^(n-1)—we calculate f'(x) = -4x + 4. This new function, f'(x), can then tell us how f(x) is behaving; where it's increasing or decreasing. For students, picturing the original function as a path and the derivative as the steepness at any moment can make this concept more tangible.

Critical Numbers

Critical numbers play a pivotal role in identifying the highs and lows of a function, metaphorically speaking. They are the x-values where the function's derivative is zero or the derivative does not exist. At these points, the slope of the tangent to the function's curve is horizontal.

In our example, after finding the derivative f'(x) = -4x + 4, we look for critical numbers by setting the derivative equal to zero, resulting in the equation -4x + 4 = 0. Solving this gives us x = 1, our sole critical number. Why are these numbers critical? Because they signify potential peaks or valleys on the graph—a place where hikers might find the view most impressive or a place to catch their breath on the function's landscape.

First Derivative Test

The First Derivative Test is the ultimate arbiter when determining the nature of critical points—whether they're the tops of hills (local maxima) or bottoms of valleys (local minima). After locating critical numbers, this test helps us understand the behavior of the function around these critical points by examining the sign of the derivative before and after.

For our given function, at the critical number x = 1, we explore the sign of the derivative on either side. If we plug in x = 0 into f'(x), we get f'(0) = 4, which is positive, indicating the function is increasing before the critical point. On the other side, with x = 2, f'(2) = -4 is negative, denoting a decrease. The switch from positive to negative as we pass x = 1 means we've crested a hill—x = 1 is a local maximum for our function. By employing this test, students can confidently determine the local high and low points on the function's journey.