09_Ex 2.9

Exercise Questions

Questions	Question Links
Q1. Determine the interval in which f is…	2.9 Q-1
Q2. Find the extreme value for the following function..	2.9 Q-2
Q3. Find the maximum and minimum value of the function…	2.9 Q-3
Q4. Show that $y=\frac{\ln x}{n}$ has maximum value at $x=e$	2.9 Q-4
Q5. Show that $y=x^{x}$ has minimum value at $x=\frac{1}{e}$.	2.9 Q-5

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Overview

This exercise focuses on analyzing the behavior of functions—whether they are increasing or decreasing—over specified intervals within their domains. It also emphasizes identifying extreme values, such as maxima, minima, and points of inflection, using first and second derivatives. These concepts are crucial for understanding the rates of change and optimizing functions in mathematical and applied contexts.

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Key Concepts

1. Increasing and Decreasing Intervals:

   • A function $f(x)$ is increasing on an interval if its derivative $f'(x) > 0$ throughout that interval.
   • A function $f(x)$ is decreasing on an interval if its derivative $f'(x) < 0$ throughout that interval.

2. Critical Points:

   • Points where $f'(x) = 0$ or $f'(x)$ is undefined. These points are candidates for extreme values.

3. Second Derivative Test:

   • Determines the concavity of the function:
     • If $f''(x) > 0$, the function is concave upward, indicating a relative minimum.
     • If $f''(x) < 0$, the function is concave downward, indicating a relative maximum.

4. Inflection Points:

   • Points where $f''(x) = 0$ and the concavity changes.

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Important Formulas

1. Derivative of Basic Functions:

   • $\frac{d}{dx}[\sin x] = \cos x$, $\frac{d}{dx}[\cos x] = -\sin x$, $\frac{d}{dx}[x^n] = n x^{n-1}$

2. Finding Critical Points:

   • Solve $f'(x) = 0$ to find candidates for maxima, minima, or inflection points.

3. Second Derivative Test:

   • Evaluate $f''(x)$ at critical points to confirm the nature of the extremum.

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Tips and Tricks

1. Symmetry in Trigonometric Functions:

   • Use the periodicity and symmetry of sine and cosine functions to simplify interval analysis.

2. Sign Charts for Derivatives:

   • Create a sign chart for $f'(x)$ to quickly determine increasing or decreasing intervals.

3. Extreme Value Theorem:

   • For continuous functions on a closed interval, check endpoints along with critical points to determine the absolute extrema.

4. Simplifying Quadratics:

   • Factorize or complete the square when analyzing quadratic functions for critical points.

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Summary

This exercise equips students with techniques to determine the behavior of functions over specified intervals. Key takeaways include:

• Using derivatives to find where a function increases, decreases, or reaches extremum points.
• Applying the second derivative to confirm the nature of critical points.
• Leveraging symmetry and algebraic manipulation to simplify complex calculations.

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Reference

By SIr Shahzad Sair:

By Great Science Academy:

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