The difference between the greatest and least values of the...

The difference between the greatest and least values of the function $f(x)=\sin 2 x-x$ on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is (a) $\pi$ (b) $\left(\frac{\sqrt{3}}{2}+\frac{\pi}{3}\right)$ (c) $\left(\frac{\sqrt{3}+\sqrt{2}}{2}+\frac{\pi}{6}\right)$ (d) $\left(\frac{\sqrt{2}+\sqrt{3}}{2}\right)$

The difference between the greatest and least values of the function $f(x)=\sin 2 x-x$ on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is
(a) $\pi$
(b) $\left(\frac{\sqrt{3}}{2}+\frac{\pi}{3}\right)$
(c) $\left(\frac{\sqrt{3}+\sqrt{2}}{2}+\frac{\pi}{6}\right)$
(d) $\left(\frac{\sqrt{2}+\sqrt{3}}{2}\right)$

Key Concepts

Range of a Function

The range of a function is the set of values that the function can take. In optimization problems, determining the difference between the maximum and minimum values calculates the spread of the function's range over the interval, which is a standard measure of variability in the function's output.

Trigonometric Functions

Trigonometric functions, like the sine function, are fundamental in studying periodic behavior in mathematics. They express relationships between angles and lengths in circles, and modifications such as scaling the input (e.g., sin 2x) change the period of the function, which is a key aspect when analyzing their behavior over an interval.

Differentiation and Critical Points

Differentiation is used to determine the rate of change of a function with respect to its variable. Finding critical points by setting the derivative equal to zero or identifying points where the derivative is undefined helps in locating local maxima and minima of the function, a fundamental step in optimization.

Optimization on Closed Intervals

Optimization on closed intervals involves finding the maximum and minimum values of a function within a given bounded domain. The Extreme Value Theorem guarantees that if a function is continuous on a closed interval, it must attain both a maximum and a minimum value, either at critical points or at the boundaries of the interval.

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