find the absolute maximum value on (0, ∞) for f(x) = 5x^6 / e^x. select the correct choice below and, if…

find the absolute maximum value on (0, ∞) for f(x) = 5x^6 / e^x. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. a. the absolute maximum is □ at x = □. (round to two decimal places as needed.) b. there is no absolute maximum.

find the absolute maximum value on (0, ∞) for f(x) = 5x^6 / e^x. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. a. the absolute maximum is □ at x = □. (round to two decimal places as needed.) b. there is no absolute maximum.

Answer

Explanation:

Step1: Find the derivative of (f(x))

Use the quotient rule ((\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here (u = 5x^{6}), (u^\prime=30x^{5}), (v = e^{x}), (v^\prime=e^{x}). Then (f^\prime(x)=\frac{30x^{5}e^{x}-5x^{6}e^{x}}{(e^{x})^{2}}=\frac{5x^{5}e^{x}(6 - x)}{e^{2x}}=\frac{5x^{5}(6 - x)}{e^{x}})

Step2: Find the critical points

Set (f^\prime(x)=0). Since (e^{x}>0) for all (x), we solve (5x^{5}(6 - x)=0). The solutions are (x = 0) and (x = 6). But (x\in(0,\infty)), so we consider (x = 6)

Step3: Use the second - derivative test or analyze the sign of (f^\prime(x))

For (x\in(0,6)), let's take a test point (x = 1). Then (f^\prime(1)=\frac{5\times1^{5}(6 - 1)}{e^{1}}=\frac{25}{e}>0). For (x\in(6,\infty)), let's take a test point (x = 7). Then (f^\prime(7)=\frac{5\times7^{5}(6 - 7)}{e^{7}}=\frac{-5\times7^{5}}{e^{7}}<0)

Step4: Find the function value at (x = 6)

(f(6)=\frac{5\times6^{6}}{e^{6}}\approx\frac{5\times46656}{403.4288}\approx578.71)

Answer:

A. The absolute maximum is (578.71) at (x = 6)