Given f(t) =t^2 − t − ln(t)

Question

a) Find the intervals on which f is increasing or decreasing,

b) Find the local maximum and minimum values of f

c) Find the interval of concavity and inflection points

Answer 1

Ethan,

the domain of the function is (0, infinity) because of the lnt.

for a) and b) we need the derivative:

f'(t) = 2t - t - 1/t

a) the derivative does not exist at 0, but it is not in the domain already.

Set f'(t) = 0, and solve: 2t² - t - 1= 0 (2t + 1)( t - 1) = 0 --> t = - 1/2 and t = 1

Only t=1 is in the domain, that is the only critical value.

If 0 < t < 1, the f'(t) < 0 and if t > 1 the f'(t) > 0, so f(t) is decreasing on (0 , 1) and

increasing on (1, infinity) and therefore t=1 is a local minimum.

b) f''(t) = 2 + 1/t² Obviously f''(t) = 0 has no solution and f''(t) > 0 on its entire domain, so f(t) is concave up.

There is no inflection point because there is no sign change in f''.

So f(t) is concave up on (0, infinity)