Mathematics

Problem

A function

f

is defined by

f(x) = \frac{3}{x^2+2}

x \in \mathbb{R}

. The region

R

is bounded by the curve

y = f(x)

, the

x

-axis and the lines

x = 0

and

x = \sqrt{6}

. Let

A

be the area of

R

. The line

x = k

divides

R

into two regions of equal area. Let

m

be the gradient of a tangent to the curve

y = f(x)

. a. Sketch the curve

y = f(x)

, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes. b. Show that

A = \frac{\sqrt{2}\pi}{2}

. c. Find the value of

k

. d. Show that

m = -\frac{6x}{(x^2+2)^2}

. e. Show that the maximum value of

m

\frac{27}{32} \sqrt{\frac{2}{3}}