5.find the concavity and inflection points if exist. 6 also find relative extrema.

Question

5. Let f(x) = (- cos x + sin x) over the interval [-,]. Use the first and second derivatives off to determine where f is increasing, decreasing, concave up, and concave down. Locate all inflection points,if they exist.

6. Find the relative extrema of f(x) =(x + 3) /(x-2)

Answer 1

∝1st : d/(dx)(sin(x) - cos(x)) = sin(x) + cos(x)

Increasing : f'(x) > 0 ⇒ Periodical intervals (-π/4 + 2πk, 3π/4+2πk)

Decreasing on the others.

2nd: (d^2)/(dx^2)(sin(x) - cos(x)) = -sin(x) + cos(x)

Concave up : f''(x) > 0 ⇒ f'(x) is increasing ⇒ Periodical intervals (-3π/4 + 2πk, π/4 + 2πk)

Concave down on the others.

6. Since it has a vertical asymptote at x = 2, check for a horizontal asymptote, which is y=1 because both powers of x, in numerator and denominator, are the same with coefficients 1 and 1, 1/1 = 1.

You have to specify intervals because this is what you get:

f'(x) = -5/(x - 2)^2 which is negative on anything but x = 2, i.e. you function always decreases.

on x∈ (-∞,2) f(x)∈ (1,-∞) and on x∈ (2,∞) f(x)∈ (∞,1)