Question

Given f(x)=cos(5x)on the closed interval [0,2π/5]
Find the open intervals where f is concave down and the open intervals where f is concave up. Express your answers in interval notation with exact values.

Answer 1

f"(x) = -25cos(5x)
f is concave up where f" is positive: (π/10,3π/10)
f is concave down on [0,π/10)U(3π/10,4π/10]

see the graph at

https://www.wolframalpha.com/input/?i=cos%285x%29

Answer 2

but what would it be within the interval though?

Answer 3

f' = -5 sin 5x
f" = -25 cos 5x
where f" is negative, concave down (sheds water)
where f" is positive, concave up (holds water)
f" is negative where cos 5x is positive
that is where 5x = 0 to 5x = pi/2 and from 5x = 3 pi/2 to 5x = 2 pi
the rest of the circle is where f" is positive

Answer 4

what are the critical values for this problem?

Answer 5

as always, where f'(x) = 0
f' = -5sin(5x)
where is sin(5x)=0 ?
Verify your answer with the graph referenced above.

Answer 6

For heavens sake graph it !