The continuous extension of (sin x)x to [0, ?]a. Graph

Chapter 4, Problem 89E

(choose chapter or problem)

The continuous extension of \((\sin x)^{x}\)  to \([0, \pi]\)

a. Graph \(f\left(x)=(\sin x)^{x}\right.\) on the interval \(0 \leq x \leq \pi\). What value would you assign to \(f\) to make it continuous at \(x=0\)?

b. Verify your conclusion in part (a) by finding \(\lim _{x \rightarrow 0^{+}} f(x)\) with l'Hôpital's Rule.

c. Returning to the graph, estimate the maximum value of \(f\) on \(f\). About where is max \(f\) taken on?

d. Sharpen your estimate in part (c) by graphing \(f^{\prime}\) in the same window to see where its graph crosses the \(x\)-axis. To simplify your work, you might want to delete the exponential factor from the expression for \(f^{\prime}\) and graph just the factor that has a zero.

Equation Transcription:

     

   

Text Transcription:

(sin x)^x

[0, pi]

fx) = (sin x)^x

0 < or = x < or = pi

f

x=0

lim_x right arrow 0^+ f(x)

F

[0, pi]

f

f’

x

f’