Problem 22
Question
Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=-(x+3)^{2}\)
Step-by-Step Solution
Verified Answer
The critical point is at \( x = -3 \) and \( f(x) \) has a local maximum there.
1Step 1: Find the Derivative
Start by finding the derivative of the function \( f(x) = -(x + 3)^2 \). To do this, apply the power rule followed by the chain rule. The derivative is \( f'(x) = -2(x + 3) \).
2Step 2: Set the Derivative to Zero
To find where \( f'(x) = 0 \), set the derivative equal to zero: \(-2(x + 3) = 0 \). Solve for \( x \) to find the critical points.
3Step 3: Solve for Critical Point
Solve the equation \( -2(x + 3) = 0 \). Divide both sides by \(-2\) to obtain \( x + 3 = 0 \). Therefore, the critical point is \( x = -3 \).
4Step 4: Determine the Nature of the Critical Point
Check the sign of \( f'(x) \) before and after \( x = -3 \) to determine the nature of the critical point. For \( x < -3 \), \( f'(x) > 0 \) and for \( x > -3 \), \( f'(x) < 0 \). This indicates \( f(x) \) changes from increasing to decreasing at \( x = -3 \).
5Step 5: Conclusion of Local Extremum
Since \( f(x) \) changes from increasing to decreasing at \( x = -3 \), it indicates a local maximum at \( x = -3 \).
Key Concepts
Understanding DerivativesDiscovering Critical PointsExploring Local ExtremaUnderstanding the Chain Rule
Understanding Derivatives
A derivative represents how a function changes as its input changes. In simpler terms, it gives us the slope of the function's curve at any given point. For a function like \( f(x) = -(x + 3)^2 \), this means checking how steeply the curve goes up or down as \( x \) changes.
To find the derivative, we use rules such as the power rule and the chain rule. The power rule helps us differentiate expressions of the form \( x^n \), where the derivative becomes \( nx^{n-1} \). In our case, we had \( (x + 3)^2 \), and applying the power rule gives us one part of the solution.
The chain rule is essential when dealing with composite functions, which are functions inside a function, like \( (x + 3)^2 \). It tells us to multiply the derivative of the outer function by the derivative of the inner function, leading to \( f'(x) = -2(x + 3) \). This result gives us the slope of \( f(x) \) at any point \( x \).
To find the derivative, we use rules such as the power rule and the chain rule. The power rule helps us differentiate expressions of the form \( x^n \), where the derivative becomes \( nx^{n-1} \). In our case, we had \( (x + 3)^2 \), and applying the power rule gives us one part of the solution.
The chain rule is essential when dealing with composite functions, which are functions inside a function, like \( (x + 3)^2 \). It tells us to multiply the derivative of the outer function by the derivative of the inner function, leading to \( f'(x) = -2(x + 3) \). This result gives us the slope of \( f(x) \) at any point \( x \).
Discovering Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are essential as they often indicate potential local minima or maxima. A critical point is where the slope of the tangent to the function becomes zero, meaning the curve is momentarily flat.
In our example, after differentiating \( f(x) \) and getting the derivative \( f'(x) = -2(x + 3) \), we find critical points by setting the derivative to zero: \(-2(x + 3) = 0\).
By solving this equation, we discover that the critical point is at \( x = -3 \). This is the point where the rate of change of our function stops, signaling a potential shift from increasing to decreasing or vice versa. This understanding is pivotal in the examination of the function's behavior.
In our example, after differentiating \( f(x) \) and getting the derivative \( f'(x) = -2(x + 3) \), we find critical points by setting the derivative to zero: \(-2(x + 3) = 0\).
By solving this equation, we discover that the critical point is at \( x = -3 \). This is the point where the rate of change of our function stops, signaling a potential shift from increasing to decreasing or vice versa. This understanding is pivotal in the examination of the function's behavior.
Exploring Local Extrema
Local extrema are the highest or lowest points in a particular section of a graph. A local maximum is a peak, while a local minimum is a trough within a neighborhood of a curve.
To identify whether a critical point is a local extremum, we need to examine the behavior of the derivative around it. In our example, at \( x = -3 \), the sign of the derivative \( f'(x) = -2(x + 3) \) tells us a lot.
To identify whether a critical point is a local extremum, we need to examine the behavior of the derivative around it. In our example, at \( x = -3 \), the sign of the derivative \( f'(x) = -2(x + 3) \) tells us a lot.
- For \( x < -3 \), \( f'(x) > 0 \), which means the function is increasing.
- For \( x > -3 \), \( f'(x) < 0 \), indicating the function is decreasing.
Understanding the Chain Rule
The chain rule is a method used to find the derivative of composite functions. These are functions made up of other functions, like \( -(x + 3)^2 \). It's vital for computing the rate of change for more complex functions.
When using the chain rule, differentiate the outer function first. Treat the inner function as a constant. Then multiply by the derivative of the inner function. For \( -(x + 3)^2 \), the outer function is \( -(u)^2 \) and the inner function is \( (x + 3) \).
When using the chain rule, differentiate the outer function first. Treat the inner function as a constant. Then multiply by the derivative of the inner function. For \( -(x + 3)^2 \), the outer function is \( -(u)^2 \) and the inner function is \( (x + 3) \).
- Differentiate the outer: \(-2u\).
- Differentiate the inner: the derivative of \( x+3 \) is 1.
- Multiply them to get \( -2(x + 3) \).
Other exercises in this chapter
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