
#Times f(x)=0
f(x) = x^{4}4x^{3}8x^{2}+4
How many times does f(x) = 0? (excuse the bad formulation)
Hint: Gather as much information from the derivative of f(x) as possible. We are only interested in the number of times f(x) equals zero.
Okay so differentiating the function is simple: f ' (x) = 4x^{3}12x^{2}16x, but what kind of information do I need to extract in order to determine how many times f(x)=0?

Re: #Times f(x)=0
Hi Cinnaman,
a few tipps:
f'(x)=0 => x1=1, x2= 0; x3=4 the event. max and min points of f(x)
If f''(x1) < 0 => x1 is max. point =>MAX1( x1f(x1))
If f''(x1) > 0 => x1 is min. point =>MIN1( x1f(x1))
analog with x2 und x3.
EXTx= Extrem xpoint;
xp<xn;
If sign of f(EXTxp) is not = f(EXTxn) => there is a x, beetwen xp und xn with f(x)=0

Re: #Times f(x)=0
Ah, that makes sense, thank you.
I figured out that there is no x where f(x)=0 between x=1 and x=0, but there is one between x=0 and x=4.
Is the last step now to take the limit as x approaches inf / inf and use them as extreme points aswell?

Re: #Times f(x)=0