Question

Identify the critical points. Then use (a) the First Derivative Test and (if possible) (b) the Second Derivative Test to decide which of the critical points give a local maximum and which give a local minimum. $$ f(x)=x^{3}-6 x^{2}+4 $$

Short answer

Critical points: \( x = 0 \) local max, \( x = 4 \) local min.

Step-by-step solution

Find the First Derivative

To find the critical points of the function, we first need to take the derivative of the function. The function is given by \( f(x) = x^3 - 6x^2 + 4 \). Find the first derivative \( f'(x) \). Using the power rule, we have: \[ f'(x) = 3x^2 - 12x. \]

Find Critical Points

To find critical points, set the first derivative equal to zero and solve for \( x \). This gives us: \[ 3x^2 - 12x = 0. \] Factor the equation: \[ 3x(x - 4) = 0. \] Therefore, \( x = 0 \) and \( x = 4 \) are the critical points.

Use the First Derivative Test

The critical points found are \( x = 0 \) and \( x = 4 \). Test intervals around these points to determine the nature of each critical point. Choose test points such as \( x = -1 \), \( x = 2 \), and \( x = 5 \).- For \( x = -1 \), \( f'(-1) = 3(-1)^2 - 12(-1) = 3 + 12 = 15 \) (positive).- For \( x = 2 \), \( f'(2) = 3(2)^2 - 12(2) = 12 - 24 = -12 \) (negative).- For \( x = 5 \), \( f'(5) = 3(5)^2 - 12(5) = 75 - 60 = 15 \) (positive).The sign changes from positive to negative at \( x = 0 \), indicating a local maximum, and from negative to positive at \( x = 4 \), indicating a local minimum.

Use the Second Derivative Test (optional)

Find the second derivative to confirm findings:\[ f''(x) = \frac{d}{dx}(3x^2 - 12x) = 6x - 12. \]Evaluate \( f''(x) \) at the critical points:- \( f''(0) = 6(0) - 12 = -12 \) (negative, confirming a local maximum).- \( f''(4) = 6(4) - 12 = 24 - 12 = 12 \) (positive, confirming a local minimum).This confirms \( x = 0 \) is a local maximum and \( x = 4 \) is a local minimum.

Key concepts

First Derivative Test

The First Derivative Test is a crucial tool in calculus to determine the nature of critical points. Critical points are values of \( x \) at which the first derivative of the function \( f'(x) \) is equal to zero or undefined.
To apply the First Derivative Test:

• Identify critical points by setting the first derivative \( f'(x) \) of a function to zero.
• Choose test points on intervals around each critical point.
• Determine the sign of \( f'(x) \) at these test points.

Changes in the sign of the derivative around the critical points tell us about local maxima and minima. If \( f'(x) \) changes from positive to negative, the function has a local maximum. If it changes from negative to positive, there is a local minimum. If there is no sign change, the test is inconclusive.

Second Derivative Test

The Second Derivative Test offers another method for determining local extrema at critical points. It often serves as a confirmation of the first derivative findings and is especially helpful in cases where the sign changes of the first derivative are tricky to interpret.
Here's how it works:

• Calculate the second derivative \( f''(x) \) of the function.
• Evaluate \( f''(x) \) at each critical point found from the first derivative.

The second derivative tells us about the concavity of the function:

• If \( f''(x) > 0 \) at a point, the function is concave up, indicating a local minimum.
• If \( f''(x) < 0 \), the function is concave down, suggesting a local maximum.
• If \( f''(x) = 0 \), the test is inconclusive, and more analysis or another method might be needed.

Local Maximum and Minimum

In calculus, determining local maxima and minima is crucial in understanding the behavior of functions. Local maximum points are where a function shifts from increasing to decreasing. Local minimum points occur where the function changes from decreasing to increasing.
These points are not necessarily the highest or lowest points on the entire graph but are the peaks and valleys observed in the vicinity of the critical points.
The First and Second Derivative Tests are primary methods to identify these local extrema. The First Derivative Test allows for analysis of intervals around critical points, while the Second Derivative Test uses concavity to confirm these points more succinctly.
Recognizing local maxima and minima helps in graphing and optimizing functions, making these concepts fundamental in both pure and applied mathematics.

Derivative Calculation

Calculating derivatives is a foundational part of calculus, serving as a tool to understand the rate of change within a function. For anyone exploring calculus, being comfortable with derivative calculation is essential.
The derivative, symbolized as \( f'(x) \) for a function \( f(x) \), represents the slope of the tangent at any point on the function's graph.
Using rules like the power rule helps simplify this process. For example, to find the first derivative of \( f(x) = x^3 - 6x^2 + 4 \), apply the power rule to determine \( f'(x) = 3x^2 - 12x \).
Finding subsequent derivatives, like the second derivative, can assist in deeper analysis of the function, as seen with \( f''(x) = 6x - 12 \). By embracing derivative calculation, one can delve into the myriad applications from determining instantaneous changes to graph analysis.