Question

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

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: \(x = \ln 6\)