Question

Please help for this question what is the function and thanks

Answer 1

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`