Question

71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]

Accepted Answer

First, understand that the function to analyze is $$f(x) = \cos(\ln x) on $$the interval $$\left[ \frac{1}{2}, 2 ight]. We $$want to find the absolute maximum and minimum values on this closed interval. Step 1: Find the derivative of the function to locate critical points. Using the chain rule, the derivative is $$f'(x) = -\sin(\ln x) \cdot \frac{1}{x}. $$Step 2: Set the derivative equal to zero to find critical points inside the interval: $$-\sin(\ln x) \cdot \frac{1}{x} = 0. $$Since $$\frac{1}{x} 
eq 0$$ for $$x \u003e 0$$, this reduces to $$\sin(\ln x) = 0. $$Step 3: Solve $$\sin(\ln x) = 0$$ for $$x in$$ $$\left[ \frac{1}{2}, 2 ight]. $$Recall that $$\sin t = 0$$ when $$t = k\pi$$ for any integer $$k. So$$, set $$\ln x = k\pi$$ and solve for $$x = e$$^{k\pi}$$, then determine which values lie in the interval. Step 4: Evaluate the function f(x$$) = \cos(\ln x)$$ at all critical points found in the interval and also at the endpoints x$$ = \frac{1}{2}$$ and x = 2$. Compare these values to identify the absolute maximum and minimum.

71. Locate and identify the absolute extreme values of cos(ln x) on [1/2, 2]

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Key Concepts

Absolute Extreme Values

Critical Points and Derivatives

Chain Rule and Differentiation of Composite Functions