Question

Find the interval of the concavity for the function f(x) = (sinx + cosx)e^x in (0, 2π.)

Answer 1

We have f(x) = (sinx + cosx)e^x

f'(x) = (sinx + cosx)e^x + e^x(cosx – sinx)

f'(x) = e^x(sinx + cosx + cosx – sinx)

f'(x) = 2e^xcosx f ≤(x) = 2 (e^xcosx – e^xsinx)

f ≤ (x) = 2e^x(cos x – sinx)

Now, f ≤ (x) = 0 gives 2e^x(cosx – sinx) = 0

⇒ tan x = 1

⇒ x = π/4 ,  5π/4

By the sign scheme for the function f"(x) = 0, we have f(x) is concave down in (π/4, 5π/4) and concave up in (0, π/4) ∪ (5π/4 , 2π).