Question 01 (i)
If the derivative of is given by
find the interval or intervals on which is concave down.
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Find the second derivative of f(x).
What do the signs of the second derivative tell you about concavity?
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We have that
The second derivative is (using quotient rule),
In order to find intervals of concavity we must check where the second derivative is zero or does not exist. We see that the second derivative is zero whenever 1-lnx=0. This occurs if x=e. The second derivative doesn't exist if x=0 (the denominator vanishes) or if x<0 since then the logarithm cannot be evaluated. In fact we anticipate that x<0 is not even in the domain of the function. Therefore our intervals of interest are . We can make a table (or number line) as follows
Therefore we have that f(x) is concave up on and concave down on .