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\na.\n(type your answer in interval notation type an exact answer. use a comma to separate answers as needed)\nb. the graph is never concave downward

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\na.\n(type your answer in interval notation type an exact answer. use a comma to separate answers as needed)\nb. the graph is never concave downward

Answer

Explanation:

Step1: Find the first derivative

Using the derivative rules ((e^{ax})^\prime = ae^{ax}), for (y = 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 derivative 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), so (24e^{x}-4e^{2x}=0). Factor out (4e^{x}): (4e^{x}(6 - e^{x})=0). Since (e^{x}>0) for all (x\in R), then (6 - e^{x}=0), which gives (e^{x}=6) and (x = \ln6).

Step4: Test intervals for concavity

  • For (x<\ln6), let (x = 0). Then (y^{\prime\prime}(0)=24e^{0}-4e^{0}=24 - 4=20>0).
  • For (x>\ln6), let (x=\ln(12)). Then (y^{\prime\prime}(\ln(12))=24e^{\ln(12)}-4e^{2\ln(12)}=24\times12-4\times144=288 - 576=- 288<0).

Answer:

  • The graph of (f(x)) is concave upward on the interval ((-\infty,\ln6)).
  • The graph of (f(x)) is concave downward on the interval ((\ln6,\infty)).