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Please help for this question what is the function and thanks

Please help for this question what is the function and thanks-example-1

1 Answer

11 votes

Answer:

b) local minimum at
x=(1)/(e)

c) Graph of the function is convex

Explanation:

We are told that
f'((1)/(e))=0, which implies there is a turning/stationary point at
x=(1)/(e).

Substituting
x=(1)/(e) into
f''(x) will tell us if the turning point is a minimum or a maximum:


f''((1)/(e))=(1)/((1)/(e) ) =e>0 \implies \textsf{local minimum}

Therefore, statement a) is false and statement b) is true.

If the function has a minimum turning point, then this implies that the curve is convex. Therefore, statement c) is true.

Extremum = local min and max points.

We have already established that there is a local minimum at
x=(1)/(e), therefore statement d) is false.

I know this is not needed for this question, but here are the workings to detemine the equation of the function (I've also attached a graph). This supports the answers above.


\textsf{if} \ f''(x)=(1)/(x)\\\\\implies f'(x)=\int f''(x) \ dx \\\\\implies f'(x)=ln|x|+C\\


\textsf{if} \ f'((1)/(e))=0\\\\\implies ln|(1)/(e)|+C=0\\\\\implies -1+C=0\\\\\implies C=1\\\\


\implies f'(x)=ln|x|+1\\


\textsf{if} \ f'(x)=ln|x|+1\\\\\implies f(x)=\int f'(x) \ dx\\\\\implies f(x)=xln(x)-x+x+C\\\\\implies f(x)=xln(x)+C

Please help for this question what is the function and thanks-example-1
User Ofstudio
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