# The productivity of a company during the day is given by # Q(t) = -t^3 + 9t^2 +12t # at time t minutes after 8 o'clock in the morning. At what time is the company most productive?

##### 1 Answer

Mar 22, 2017

2:36 pm

#### Explanation:

The productivity is given as:

# Q(t) = -t^3 + 9t^2 +12t #

To find the optimum productivity we seek a critical point of

Differentiating wrt

# (dQ)/dt = -3t^2 + 18t +12 #

At a critical point

# -3t^2 + 18t^ +12 = 0 #

# :. t^2 -6t^ -4 = 0 #

# :. t=3+-sqrt(13) #

We require

We can do a second derivative test to verify this is a maximum;

# (d^2Q)/dt^2 = -6t + 18 #

When

Thus the maximum productivity occurs when

ie,

As