find the critical numbers of the function. (enter your answers as a comma - separated list. if an answer…

find the critical numbers of the function. (enter your answers as a comma - separated list. if an answer does not exist, enter dne.)\n\n$f(x)=x^{4}e^{-5x}$\n\n$x=$\n\nresources\n\nread itwatch it

find the critical numbers of the function. (enter your answers as a comma - separated list. if an answer does not exist, enter dne.)\n\n$f(x)=x^{4}e^{-5x}$\n\n$x=$\n\nresources\n\nread itwatch it

Answer

Explanation:

Step1: Differentiate the function using the product rule

The product rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Let (u = x^{4}) and (v=e^{-5x}). (u^\prime = 4x^{3}) (using the power rule ((x^{n})^\prime=nx^{n - 1})), (v^\prime=- 5e^{-5x}) (using the chain rule ((e^{ax})^\prime=ae^{ax})) (f^\prime(x)=4x^{3}e^{-5x}+x^{4}(-5)e^{-5x}=x^{3}e^{-5x}(4 - 5x))

Step2: Set the derivative equal to zero

Since (e^{-5x}>0) for all (x\in R), we set (x^{3}(4 - 5x)=0) (x^{3}=0) gives (x = 0) (4-5x=0) gives (x=\frac{4}{5})

Answer:

(0,\frac{4}{5})