Yes, you are correct. To find the values of t when the velocity is increasing for the given position function s(t) = t⁴ - 4t³, we need to analyze the acceleration (the second derivative) of the function.
The velocity function is obtained by taking the derivative of the position function with respect to time:
v(t) = d(s(t))/dt = d(t⁴ - 4t³)/dt
Taking the derivative, we get:
v(t) = 4t³ - 12t²
To find when the velocity is increasing, we need to examine the sign of the acceleration. The acceleration function is obtained by taking the derivative of the velocity function:
a(t) = d(v(t))/dt = d(4t³ - 12t²)/dt
Taking the derivative, we get:
a(t) = 12t² - 24t
To determine when the acceleration is positive (indicating increasing velocity), we set a(t) > 0 and solve for t:
12t² - 24t > 0
Factoring out a common factor of 12t:
12t(t - 2) > 0
Now we have two factors to consider: 12t and (t - 2). To determine the intervals where the inequality is true, we can use a sign chart or analyze each factor separately:
Factor 12t: For 12t > 0, t must be greater than 0 (t > 0).
Factor (t - 2): For (t - 2) > 0, t must be greater than 2 (t > 2).
Combining the intervals, we find that the velocity is increasing for t > 2. In other words, the velocity is increasing after t = 2.
Therefore, the values of t when the velocity is increasing for the given position function are t > 2.