. (a) Make a conjecture about the shape of the graph ofy = 12 x ln x, and draw a rough

Chapter 3, Problem 37

(choose chapter or problem)

(a) Make a conjecture about the shape of the graph of \(y=\frac{1}{2} x-\ln x\), and draw a rough sketch.

(b) Check your conjecture by graphing the equation over the interval \(0<x<5\) with a graphing utility.

(c) Show that the slopes of the tangent lines to the curve at \(x=1\) and \(x=e\) have opposite signs.

(d) What does part (c) imply about the existence of a horizontal tangent line to the curve? Explain.

(e) Find the exact x-coordinates of all horizontal tangent lines to the curve.

Equation Transcription:

Text Transcription:

y=1/2 x−ln x

0<x<5

x=1

x=e