Find $ f $.
$ f'(x) = 5x^4 - 3x^2 + 4 $, $ \quad f(-1) = 2 $
Okay we need to find a function F of X. Such that f of negative one is two and the derivative F prime of x is five X to the fourth minus three X squared plus four. Ffx will be uh the anti derivative or the indefinite integral of F prime of X. So f of X equals the integral of F prime of X. Uh That will be the integral of five X to the fourth -3 x squared plus four D X. We're integrating with respect to X Integral of five extra fourth will be five X to the fifth over five, which will be just X to the fifth minus. Uh The integral of three, X squared will be three X cubed over three or just X squared. And Andy plus And then the integral of four will simply be four x plus some constant C. Um And we don't know what C is but because f of negative one has to equal to. That's actually going to help us find out what C is. So F of X is equal to X to the fifth minus X squared plus four X plus. Safe move this down slightly. Now keep in mind That f of -1 has to equal to. So f of negative one has to equal to But f of X equals this expression. So f of negative one, substitute negative one everywhere you see X X to the fifth is negative one to the fifth minus x squared X is negative one. So minus negative one squared Plus four times x plus four times negative one plus our constancy that we got from our integration. Uh and this has to come out equal to two because f of -1 has to equal to. So we have a little equation that we're gonna solve for C. And once we find see um then uh we will actually be done. Okay negative one to the fifth is negative one minus negative one to the second. Negative one to the second is positive one, sober, subtracting one plus four times negative one. So we're adding negative four. Or subtracting for uh last but not least plus C. Has to equal to negative one. Subtract one is negative to subtract four is negative six negative six Plus Our Constant C. Equals two. We'll see has to be eight because negative six plus eight equal to so C equals positive eight. And so we can go back up here to our function. Okay F of X. Let's put a nice little box around this. Okay so F of X equals exited fifth minus X squared plus four X plus C. And we just found below here that C has to be eight. Um So let's just right at sea, We now know is eight. So our function F. Of X is equal to X to the fifth minus X squared plus four X plus eight.