3. Let f be the function defined by f(x)=ln(2+sinx) for pi<=x<=2pi

Question

a. Find the absolute maximum value and the absolute minimum value of f. Show the analysis that leads to your conclusion.
b. Find the x-coordinate of each inflection point on the graph of f. Justify your answer.

Answer 1

f'(x) = -cosx/(2+sinx)
for max/min,
-cosx/(2+sinx)= 0
cosx = 0
x = π/2 or x = 3π/2
when x =π/2
f(π/2) = ln(2 + 1) = ln 3
when x = 3π/2
f(3π/2) = ln(2-1) = ln 1 = 0
so which is max and which is min ?

for pts of inflection, find
f''(x), then set that equal to zero
I suggest the quotient rule.