# B Do all the peaks and valleys of f have f'=0

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1. Apr 22, 2017

### AMan24

I learned in the earlier chapters that peaks and valleys of a fxn have points where f'=0 (i marked them with red x). A few chapters later it said if a fxn has 2 roots, then f'=0 (still the 1st graph).

So does that mean if the graph of a fxn is like the 2nd graph, the peaks and valleys are not f'=0? I drew where i would assume f'=0 with blue circles

2. Apr 22, 2017

If there is a knotch then f'(x) is not equal to zero......

3. Apr 22, 2017

### SlowThinker

All peaks and all valleys have f'(x)=0 but there can be a point where f'(x)=0 but it's not peak or valley. For example f(x)=x^3 at x=0.

The correct wording is that if a function has a double root, not two roots, then f'(x)=0 at that point. Happens for example with f(x)=x^2 at x=0.

4. Apr 22, 2017

Take the case y=-|x| at x=0

5. Apr 22, 2017

### SlowThinker

Yes the rules only apply if the function has a continuous derivative.
With non-continuous functions or derivatives, pretty much anything can happen.

6. Apr 26, 2017

### JonnyG

What it should say, or maybe what you meant to say, was that if a function has two roots, say $x_1$ and $x_2$, then there is a point $c$ between $x_1$ and $x_2$ such that $f'(c) = 0$