Find f(x) if `f'(x)=-6+1/x` and `f(1)=-1` .  

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lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

Sorry for the slip.

Take note of this correction.


Plug-in x=1 and f(x)= -1  to f(x)=-6x+lnx+C

`-1=-6(1)+ln1+C`

`-1=-6+0+C`         

`-1=-6+C`

`5=C`

Then, substitute the value of C to f(x).

`f(x)=-6x+lnx+C`

`f(x)=-6x+lnx+5`

`f(x)=lnx-6x+5`

Hence, the function is `f(x)=lnx-6x+5` .

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lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

To determine f(x), take the integral f'(x).

`f(x)= int f'(x)dx`

`f(x)=int(-6+1/x)dx = -int 6dx + int 1/x dx`

`f(x)=-6x + lnx + C`

Then, use the given f(1)=-1 to get the value of C.

So, plug-in x=1 and f(x)=-1.

`-1= -6(1)+ln1+C`

`-1=-6+0+C`

`-1=-6+C`

`5=C`

Plug-in the value of C to f(x).

`f(x)=-6x+ ln x + C`

`f(x)= -6x + lnx +5`

`f(x)=lnx - 6x +5`

Hence, the function is `f(x)=lnx-6x+5` .

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