Math. question?(Serious answers please)?

Question

1. Determine when the function is increasing and when it is decreasing.
(a). f(x) = In (1 + x^2)
(b).f(x) = xe^(-x).

Answer 1

Get the derivative:
1. f'(x) = 1/(1+x²) * (2x)
.... Now the first factor is always positive.
.... The second factor will change its sign at zero.
if x < 0 ........ f' < 0 , (decreasing)
if x > 0 ........ f' > 0 , (increasing)

2. f'(x) = e^(-x) + x*(-e^(-x)) = e^(-x) - xe^(-x) = (1-x)e^(-x)
.... This time e^(-x) is always positive.
.... (1-x) will change sign at 1.
if x < 1..... f' > 0
if x > 1..... f' < 0
draw your conclusion.

d:

Answer 2

f(x) = In (1 + x^2)

f'(x)=2x/(1+x^2) >0

when x>0, the function is increasing

otherwize decreasing

(b).f(x) = xe^(-x)
f'(x)=1e^(-x)+(-1)xe^(-x) >0
when x<1, the function is increasing
otherwise decreasing

Answer 3

If a function is increasing, it has a positive slope, and vice versa, so differentiate...

a. f'(x) = 2x / (1 + x^2)

The denominator must be positive, so f'(x) is negative (f(x) is decreasing) for all negative values of x, positive (f(x) is increasing) for positive values of x, and horizontal at x = 0.

b. f'(x) = x (-1) e^(-x) + e^(-x)
= (1 - x ) e^(-x)

Since e^(-x) is always positive, f'(x) will be positive (f(x) is increasing) for x less than 1, negative (f(x) is decreasing) for x greater than 1, and horizontal at x = 1.