Is the function concave up or down if

fernandoval3sbr

fernandoval3sbr

Answered question

2022-04-16

Is the function concave up or down if f(x)=(lnx)2?

Answer & Explanation

enchantsyseq

enchantsyseq

Beginner2022-04-17Added 19 answers

Step 1
It is concave up on the interval (0, e) and concave down on (e,).
f(x)=(lnx)2.
f(x)=2(lnx)1x=2lnxx.
fx)=(2x)x2lnx1x2=2(1lnx)x2
1lnx=0 where lnx=1 which is at x=e.
Step 2
All other factors of f"(x) are always positive, so the sign of f"(x) is the same as the sign of 1-lnx.
Therefore, f"(x) is positive if x<e( so lnx<1) and it is negative for x>e (where lnx>1).

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