Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) p(t)=te⁶ᵗ

Answer 1

The critical number for the function p(t)=te^6t is t = -1/6, which is found by setting the derivative of the function equal to zero and solving for t.

Step-by-step explanation:

To find the critical numbers of the function p(t)=te6t, we need to calculate the derivative of the function and find the values of t for which the derivative is either zero or undefined. The critical numbers can occur at these points. Applying product and exponential rules, let's find p'(t):

• Derivative of t is 1.
• Derivative of e6t is e6t× 6, by the chain rule.

Thus, p'(t) = e6t + te6t× 6. To find critical points, solve when p'(t) = 0:

1. e%6t(1 + 6t) = 0.
2. Since e6t is never zero, set 1 + 6t = 0.
3. Thus, t = -1/6.

The critical number of the function is therefore t = -1/6. There are no other critical numbers since the exponential function doesn't have any zeroes. As per conventions, we express our answers to problems in this section to the correct number of significant figures and proper units.