1. **State the problem:** Find the critical numbers of the function $$p(t) = t e^{5t}$$. Critical numbers occur where the derivative is zero or undefined. 2. **Find the derivative:** Use the product rule: if $$p(t) = u(t)v(t)$$, then $$p'(t) = u'(t)v(t) + u(t)v'(t)$$. Here, $$u(t) = t$$ and $$v(t) = e^{5t}$$. Calculate derivatives: $$u'(t) = 1$$ $$v'(t) = 5 e^{5t}$$ So, $$p'(t) = 1 imes e^{5t} + t imes 5 e^{5t} = e^{5t} + 5t e^{5t} = e^{5t}(1 + 5t)$$ 3. **Set the derivative equal to zero:** $$p'(t) = e^{5t}(1 + 5t) = 0$$ Since $$e^{5t}$$ is never zero, solve: $$1 + 5t = 0$$ 4. **Solve for $$t$$:** $$5t = -1$$ $$t = -\frac{1}{5}$$ 5. **Check for points where derivative is undefined:** The derivative is defined for all real $$t$$, so no other critical numbers. **Final answer:** The critical number is $$t = -\frac{1}{5}$$.