Question

find all extrema in the interval [0, 2pi] if y=sinx + cosx

and also

a certain item sells for $30. if the cost of producing this item is given by C=.05^3 + 100, find the mariginal profit when x=10.

Answer 1

To find the extrema, do the First Derivative Test.

y = sinx + cos x
y'= cosx - sinx
The extrema always occur when the derivative is zero, so set it equal to zero.

y' = cosx - sinx = 0
= cosx = sinx
cosx/cosx = sinx/cosx
1 = tanx
x = pi/4, 5pi/4

Make a sign chart...
y' increases on (0, pi/4) and (5pi/4, 2pi)
y' decreases on (pi/4, 5pi/4)
A max occurs at pi/4, a min occurs at 5pi/4.

The extrema are...
Relative maximum = √2 @ x = pi/4
Relative minimum = -√2 @ x = 5pi/4


I think you've made a mistake on your second problem. There is no variable in the C function.

Answer 2

C=.05x^3+100