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:

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

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:

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

Second Derivative Test:
 Critical Points: Points where ${f}^{\mathrm{\prime}}(x)=0$ or ${f}^{\mathrm{\prime}}(x)$ is undefined.
 Use of Second Derivative:
 If ${f}^{\mathrm{\prime}\mathrm{\prime}}(c)>0$, $c$is a local minimum.
 If ${f}^{\mathrm{\prime}\mathrm{\prime}}(c)<0$, $c$ is a local maximum.
 If ${f}^{\mathrm{\prime}\mathrm{\prime}}(c)=0$, the test is inconclusive.
Steps to Find Maxima and Minima:

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

Classify Critical Points:
 Use the first or second derivative test to classify critical points as maxima, minima, or saddle points.

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 $f(x)={x}^{3}3{x}^{2}+2$.
Solution Steps:

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

Find Critical Points by Solving ${f}^{\mathrm{\prime}}(x)=0$: $3{x}^{2}6x=0$ $3x(x2)=0$ $x=0$ or $x=2$

Analyze the Function Around Critical Points:
 Calculate ${f}^{\mathrm{\prime}\mathrm{\prime}}(x)$ to determine concavity. ${f}^{\mathrm{\prime}\mathrm{\prime}}(x)=\frac{{d}^{2}}{d{x}^{2}}(3{x}^{2}6x)$ ${f}^{\mathrm{\prime}\mathrm{\prime}}(x)=6x6$

Evaluate ${f}^{\mathrm{\prime}\mathrm{\prime}}(x)$ at Critical Points:
 ${f}^{\mathrm{\prime}\mathrm{\prime}}(0)=6(0)6=6$ (negative, indicating a local maximum)
 ${f}^{\mathrm{\prime}\mathrm{\prime}}(2)=6(2)6=6$ (positive, indicating a local minimum)

Determine the Maxima and Minima:
 At $x=0$:
 $f(0)={0}^{3}3(0{)}^{2}+2=2$ (local maximum)
 At $x=2$:
 $f(2)={2}^{3}3(2{)}^{2}+2=2$ (local minimum)
 At $x=0$:
Conclusion:
The function $f(x)={x}^{3}3{x}^{2}+2$ 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 $f(x)={x}^{3}3{x}^{2}+2$ 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.