Let be a function defined for that satisfies the differential equation dy dx = x x…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

y

be a function defined for

x>0

that satisfies the differential equation

\frac{dy}{dx}=\frac{\ln x}{x}

with initial condition

y(1)=0

. (a) Express

y(x)

as a definite integral with a constant lower limit. (b) Evaluate the integral in part (a) to obtain an explicit formula for

y(x)

. (c) Determine all intervals on

(0,\infty)

where

y

is increasing and all intervals where

y

is decreasing. Provide a clear justification based on your formula from part (b). (d) On the closed interval

[1,e^{2}]

, determine the absolute maximum and absolute minimum values of

y

. Give exact values and state where they occur.