Analysis of daily output of a factory shows that on average...

Analysis of daily output of a factory shows that, on average, the number of units per hour y produced after t hours of production is y = 24t + 0.5t2 − t3, 0 ≤ t ≤ 5. (a) Find the critical values of this function. (Assume −∞ < t < ∞. Enter your answers as a comma-separated list.) t = (b) Which critical values make sense in this particular problem? (Enter your answers as a comma-separated list.) t = (c) For which values of t, for 0 ≤ t ≤ 5, is y increasing? (Enter your answer using interval notation.)

Transcript

00:01 In this question, the amount of units produced in a factory after t hours is given by the equation y equals to 24t plus 0 .5 t squared minus t cube, where t varies between 0 and 5.

00:16 Where is to find the critical values of this function? where has to find, to determine which critical values make sense, and determine the intervals when y is increasing.

00:28 All right, let's answer the first question.

00:31 The critical values.

00:33 To find the critical values, we need to solve the equation y prime equals to 0.

00:46 In our case, y prime equals to 24 plus 0 .5 multiplied by 2 t minus 3 t squared.

01:04 This simplifies to 24 plus t minus 3 t squared.

01:13 And we want this to be equal to 0.

01:15 This gives us a quadratic equation, negative 3 t squared plus t plus 24 equals to 0 and let's find the roots of this equation by using the quadratic formula we first need to calculate the discriminant which equals to 1 squared minus 4 multiplied by negative 3 multiplied by 24 this equals to 1 plus 12 times 24 24 times 12 equals to 240 plus 48 is 288 so we are going to get 1 plus 288 this equals to 289 therefore the square root of the discriminant equals to 17 then by the quadratic formula the roots are negative 1 plus minus the square root of the discriminant divided by twice the coefficient in front of t squared and therefore the roots are 17 minus 1 is 16 and 16 divided by negative 6 is negative 16 over 6 equals to negative 8 thirds.

02:39 The second root is negative 1 minus 17 over negative 6 equals to 3.

02:47 Alright, these are the critical values of our function.

02:59 Now let's answer the second question.

03:03 Which of the critical values make sense? now recall that the critical values in our...

03:10 T, in our case, the variable t represents time.

03:14 And since t represents time, time cannot be negative, which means that t equals to negative 8 thirds does make sense.

03:48 Therefore, only t equals to 3 makes sense in this question.

04:03 Now let's answer the last question.

04:06 We are asked to find the intervals where y is increasing...