# Maxima and Minima

### Maxima and Minima in Calculus:

• Maxima and minima refer to the highest (maxima) and lowest (minima) points of a function, respectively.

### Types of Extrema:

1. Absolute Maxima and Minima:

• The highest and lowest values of a function over its entire domain.
• Represented by $f\left(c\right)$, where $c$ is the critical point or an endpoint of the domain.
2. Local Maxima and Minima:

• Points where a function reaches a high (local maxima) or low (local minima) relative to its nearby points.
• Found at critical points where the derivative equals zero or is undefined.

### Derivative Test for Extrema:

1. First Derivative Test:

• Critical Points: Points where ${f}^{\mathrm{\prime }}\left(x\right)=0$ or ${f}^{\mathrm{\prime }}\left(x\right)$ is undefined.
• Behavior around Critical Points:
• If ${f}^{\mathrm{\prime }}\left(x\right)>0$ for $x$ near a critical point $c$, $f$ is increasing around $c$ (potential local minima).
• If ${f}^{\mathrm{\prime }}\left(x\right)<0$ for $x$ near a critical point $c$, $f$ is decreasing around $c$ (potential local maxima).
2. Second Derivative Test:

• Critical Points: Points where ${f}^{\mathrm{\prime }}\left(x\right)=0$ or ${f}^{\mathrm{\prime }}\left(x\right)$ is undefined.
• Use of Second Derivative:
• If ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(c\right)>0$, $c$ is a local minimum.
• If ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(c\right)<0$, $c$ is a local maximum.
• If ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(c\right)=0$, the test is inconclusive.

### Steps to Find Maxima and Minima:

1. Find Critical Points:

• Set ${f}^{\mathrm{\prime }}\left(x\right)=0$ and solve for $x$ to find critical points.
2. Classify Critical Points:

• Use the first or second derivative test to classify critical points as maxima, minima, or saddle points.
3. Check Endpoints:

• Evaluate the function at the endpoints of the domain, if applicable, to determine absolute extrema.

### Special Cases:

• Global Maxima and Minima:
• Absolute extrema refer to the highest and lowest points of a function over its entire domain.
• Closed Intervals:
• For functions defined on closed intervals, check the values of the function at the endpoints as they might contain extrema.

### Example Problem:

Find the local maxima and minima of the function $f\left(x\right)={x}^{3}-3{x}^{2}+2$.

### Solution Steps:

1. Find the First Derivative ${f}^{\mathrm{\prime }}\left(x\right)$: ${f}^{\mathrm{\prime }}\left(x\right)=\frac{d}{dx}\left({x}^{3}-3{x}^{2}+2\right)$ ${f}^{\mathrm{\prime }}\left(x\right)=3{x}^{2}-6x$

2. Find Critical Points by Solving ${f}^{\mathrm{\prime }}\left(x\right)=0$: $3{x}^{2}-6x=0$ $3x\left(x-2\right)=0$ $x=0$ or $x=2$

3. Analyze the Function Around Critical Points:

• Calculate ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)$ to determine concavity. ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)=\frac{{d}^{2}}{d{x}^{2}}\left(3{x}^{2}-6x\right)$ ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)=6x-6$
4. Evaluate ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(x\right)$ at Critical Points:

• ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(0\right)=6\left(0\right)-6=-6$ (negative, indicating a local maximum)
• ${f}^{\mathrm{\prime }\mathrm{\prime }}\left(2\right)=6\left(2\right)-6=6$ (positive, indicating a local minimum)
5. Determine the Maxima and Minima:

• At $x=0$:
• $f\left(0\right)={0}^{3}-3\left(0{\right)}^{2}+2=2$ (local maximum)
• At $x=2$:
• $f\left(2\right)={2}^{3}-3\left(2{\right)}^{2}+2=-2$ (local minimum)

### Conclusion:

The function $f\left(x\right)={x}^{3}-3{x}^{2}+2$ has a local maximum at $x=0$ with a value of $2$ and a local minimum at $x=2$ with a value of $-2$.

### Graphical Representation:

The graph of the function $f\left(x\right)={x}^{3}-3{x}^{2}+2$ would show a local maximum at $x=0$ and a local minimum at $x=2$ based on the analysis.

### Applications:

• Optimization Problems:

• Maxima and minima concepts are essential in solving optimization problems, such as maximizing profit or minimizing cost in economics.
• Curve Sketching:

• Understanding extrema aids in accurately sketching curves, identifying peaks and valleys of a function.

### Key Considerations:

• Critical Points: Derivative equals zero or undefined.
• Test for Extrema: First or second derivative test to classify critical points.

### Limitations:

• Local vs. Global Extrema: A function might have several local extrema without possessing a global maximum or minimum.

• Singular Points: Functions might have extrema at singular points where derivatives do not exist.