Concave up or down Calculus 1 question

Question

Consider the function 𝑓(𝑥)=2(sin𝑥+cos𝑥) with domain 𝑥∈(0,𝜋). Find the interval over which the function is concave up.

Answer 1

f(x)=2(sinx + cosx)

f'(x)=2(cosx - sinx)

f''(x)=2(-sinx - cosx) (Second derivative test)

=-2(sinx + cosx)

f''(x)=0

-2(sinx + cosx)=0

sinx + cosx=0

sinx =-cosx

tanx=-1

x=3π/4 (0,π) (Inflection point)

x : (0,3π/4) f''(x) < 0 (Take a test point like π/2 to check if f''(x) is + or -)

x: (3π/4,π) f''(x) > 0 (Take a test point like 5π/6 to check if f''(x) is + or -)

Concave up interval : (3π/4,π)

Answer 2

f'(x) =2(cosx - sinx); f''(x) = 2(- sinx - cosx) > 0; sinx + cosx < 0; sinx > 0; 1 + cotx < 0; cotx < - 1;

so interval where graph is concave up is (3π/4, π)