find the intervals on which the graph of ( f ) is concave upward, the intervals on which the graph of ( f )…

find the intervals on which the graph of ( f ) is concave upward, the intervals on which the graph of ( f ) is concave downward, and the inflection points.\n( f(x)=24 e^{x}-e^{2 x} )\nfor what interval(s) of ( x ) is the graph of ( f ) concave upward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n( (-infty, ln 6) )\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nthe graph is never concave upward.\nfor what interval(s) of ( x ) is the graph of ( f ) concave downward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n( (ln 6, infty) )\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed)\nthe graph is never concave downward.\nwhat are the inflection point(s) of ( f )? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. ( x= ) (type an exact answer. use a comma to separate answers as needed.)\nb. there are no inflection points

find the intervals on which the graph of ( f ) is concave upward, the intervals on which the graph of ( f ) is concave downward, and the inflection points.\n( f(x)=24 e^{x}-e^{2 x} )\nfor what interval(s) of ( x ) is the graph of ( f ) concave upward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n( (-infty, ln 6) )\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nthe graph is never concave upward.\nfor what interval(s) of ( x ) is the graph of ( f ) concave downward? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n( (ln 6, infty) )\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed)\nthe graph is never concave downward.\nwhat are the inflection point(s) of ( f )? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. ( x= ) (type an exact answer. use a comma to separate answers as needed.)\nb. there are no inflection points

Answer

Explanation:

Step1: Find the first derivative

Using the derivative rules ((e^{ax})^\prime = ae^{ax}), for (y = f(x)=24e^{x}-e^{2x}), the first derivative (y^\prime=f^\prime(x)=24e^{x}-2e^{2x}).

Step2: Find the second derivative

Differentiate (y^\prime = 24e^{x}-2e^{2x}) again. Using the same rule, (y^{\prime\prime}=f^{\prime\prime}(x)=24e^{x}-4e^{2x}).

Step3: Set the second derivative equal to zero

Let (y^{\prime\prime}=0), then (24e^{x}-4e^{2x}=0). Factor out (4e^{x}): (4e^{x}(6 - e^{x})=0). Since (e^{x}>0) for all (x\in R), we solve (6 - e^{x}=0), which gives (e^{x}=6), and (x = \ln 6).

Step4: Test intervals for concavity

  • For (x<\ln 6), let (x = 0). Then (y^{\prime\prime}(0)=24e^{0}-4e^{0}=24 - 4=20>0). So the function is concave upward on ((-\infty,\ln 6)).
  • For (x>\ln 6), let (x=\ln 7). Then (y^{\prime\prime}(\ln 7)=24e^{\ln 7}-4e^{2\ln 7}=24\times7-4\times49=168 - 196=-28<0). So the function is concave downward on ((\ln 6,\infty)).

Answer:

  • Concave upward: ((-\infty,\ln 6))
  • Concave downward: ((\ln 6,\infty))
  • Inflection point: (x = \ln 6)