Thread: #Times f(x)=0

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!

How many times does ?

Hint: Gather as much information from the derivative of as possible.
We are only interested in the number of times f(x) equals zero.

Set and solve.

. .


There are horizontal tangents at:


The Second Derivative Test give us: .

The graph looks like this.

Code:

               |
      *        |
              -*-
       *     * | *          *
        *   *  |  *
         -*-   |
  -------------+---♥-------♥---
               |
               |    *     *
               |     *   *
               |      -*-
               |

Therefore, the graph crosses the x-axis twice.