Mathematics

Problem

Let

s(x)=\ln(x)+\dfrac{x^{2}}{4}

for

1\le x\le 3

. a. Compute

s'(x)

and determine its sign on the interval

(1,3)

. b. Using your conclusion from part (a), describe the behavior of

s

[1,3]

(increasing, decreasing, or constant). c. Determine the number of solutions to the equation

\ln(x)+\dfrac{x^{2}}{4}=2

in the interval

[1,3]

. Provide full justification for your answer.