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\n( f(x)=24 e^{x}-e^{2 x} )\n\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\na.\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\n\nb. the graph is never concave upward.

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\n( f(x)=24 e^{x}-e^{2 x} )\n\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\na.\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\n\nb. the graph is never concave upward.

Answer

Explanation:

Step1: Find the first derivative

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

Step2: Find the second derivative

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

Step3: Set the second derivative greater than 0 to find concave - upward intervals

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), we solve (6 - e^{x}>0). (e^{x}<6). Taking the natural logarithm of both sides ((y = \ln x) is an increasing function), we get (x<\ln6).

Answer:

A. ((-\infty,\ln6))