Match the difference of greatest and least value of the...

Match the difference of greatest and least value of the functions $f$ on L.H.S (a) $\sin x \sin 2 x$ on $(-\infty, \infty)$ (b) $2 x^{3}-3 x^{2}-12 x+1$ on $[-2,5 / 2]$ (c) $\cos ^{-1} x^{2}$ on $[-\sqrt{2} / 2, \sqrt{2} / 2]$ (d) $x+\sqrt{x}$ on $[0,4]$ (p) $\pi 6$ (q) $8 / 3 \sqrt{3}$ (r) 6 (s) 27

Match the difference of greatest and least value of the functions $f$ on L.H.S
(a) $\sin x \sin 2 x$ on $(-\infty, \infty)$
(b) $2 x^{3}-3 x^{2}-12 x+1$
on $[-2,5 / 2]$
(c) $\cos ^{-1} x^{2}$ on $[-\sqrt{2} / 2, \sqrt{2} / 2]$
(d) $x+\sqrt{x}$ on $[0,4]$

(p) $\pi 6$
(q) $8 / 3 \sqrt{3}$
(r) 6
(s) 27

Key Concepts

Extreme Value Analysis

This concept involves finding the maximum and minimum values of a function over a specified domain. It typically requires evaluating the function at critical points (found by setting the derivative to zero or identifying points where the derivative does not exist) and at the endpoints of the interval if it is closed. Understanding extreme value analysis is crucial for determining the difference between the greatest and least values of a function.

Domain and Range Considerations

A key aspect of analyzing any function is understanding its domain (the set of input values for which the function is defined) and its range (the set of possible output values). For problems involving extreme values, especially on closed intervals, careful consideration of the domain ensures that all potential candidates for maximum and minimum values are assessed.

Differentiation and Critical Points

Utilizing calculus, particularly the process of differentiation, helps identify critical points where the function’s rate of change is zero or undefined. These points are candidates for local extrema, and when combined with endpoint evaluations in a closed interval, they allow one to determine the absolute maximum and minimum values of a function.

Properties of Trigonometric Functions

Understanding the behavior of trigonometric functions, including their periodicity, symmetry, and bounded nature, is essential when these functions appear in a problem. Knowledge of how trigonometric and inverse trigonometric functions behave can simplify the process of finding their extreme values.

Analysis of Algebraic Functions

For algebraic functions, including polynomials and functions involving radicals, it is important to understand their continuity, differentiability, and overall behavior. Analyzing these functions often involves inspecting their derivatives, factoring, or using other algebraic techniques to find critical points and evaluate endpoints for extreme values.

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