find the following questions

Question

Consider the function f(x) = e^x /4+e^x . compute the derivative of this function

f'(x) =

a) f(x) is increasing on

b) f(x) is decreasing on

c) f(x) has a local minimum

d) f(x) has a local maximum

e) f''(x)

f) f(x) concave up

g) f(x) concave down

h)f(x) point of inflection

Answer 1

f(x) = e^x / (4 + e^x)

f'(x) = [e^x(4 + e^x) - e^x(0 + e^x)] / (4 + e^x)² = 4e^x / (4 + e^x)² > 0 for all x.

f is increasing on (-∞, ∞)

f is never decreasing

f has no local extrema

f"(x) = [4e^x(4 + e^x)² - 2(4 + e^x)(e^x)(4e^x)] / (4 + e^x)⁴ = [4e^x(4 + e^x)(4 + e^x - 2e^x)] / (4 + e^x)⁴

= [4e^x(4 - e^x)] / (4 + e^x)³ = 0 when e^x = 4. So x = ln4.

When x < ln4, f"(x) > 0. So, f is concave up

When x > ln4, f"(x) < 0. So, f is concave down

Inflection point: (ln4, f(ln4)) = (ln4, 1/2)