Question

Consider the function g(x) = sinxcosx.

a. Find an equation of the tangent line to the graph of g at (pi/3, sqrt3/4).

b. Find the critical number(s) of g on the interval [0, 2pi]. Does the function have a relative minimum, relative maximum, or neither at each critical number.

My answer:
The equation of the tangent line is y - sqrt3/4 = -1/2(x - pi/3).

The absolute minimum is at 3pi/4 and 7pi/4. The absolute maximum is at pi/4 and 5pi/4. So neither of these are relative minima/maxima.

Answer 1

g(x) = sinxcosx
g ' (x) = sinx(-sinx) + cosxcosx
= cos^2 x - sin^2 x
= cos 2x

at x = ?/3
g '(x) = cos (2?/3) = -.5
equation: y - ?3/4 = (-1/2)(x - ?/3)
So I agree with your answer.

for a max/min of g(x) , g '(x) = 0
cos 2x = 0
2x = ?/2 or 2x = 3?/2
x = ?/4 or x = 3?/4
the matching y = sin(?/4)cos(?/4)
= ?2/2*?2/2 = 1/2
or
y = sin(3?/4)(cos(3?/4))
= ?2/2 * (-?2/2)
= -1/2

check:
http://www.wolframalpha.com/input/?i=plot+y+%3D+sinxcosx+for+x%3D0+to+6.28

looking good