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f(x) = x4-4x3-8x2+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) = 4x3-12x2-16x, but what kind of information do I need to extract in order to determine how many 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( x1|f(x1))
If f''(x1) > 0 => x1 is min. point =>MIN1( x1|f(x1))
analog with x2 und x3.
EXTx= Extrem x-point;
xp<xn;
If sign of f(EXTxp) is not = f(EXTxn) => there is a x, beetwen xp und xn with 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?
Hello, Cinnaman!
Quote:
How many times does?
Hint: Gather as much information from the derivative ofas possible.
We are only interested in the number of times f(x) equals zero.
Setand solve.
. .
There are horizontal tangents at:
The Second Derivative Test give us: .
The graph looks like this.
Therefore, the graph crosses the x-axis twice.Code:|
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-------------+---♥-------♥---
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