# Double root

Gold Member

## Homework Statement

What value of ##p## gives a double root for ##f(x;p) = \cos x - 0.8+px^2##? I'm using python.

## Homework Equations

Nothing comes to mind.

## The Attempt at a Solution

I was thinking about choosing a window ##p\in[a,b]## such that ##a=0.3## yields 4 roots and ##b=0.4## yields 0 roots. Then cut this interval in half at ##p = b-0.5(a+b)## and evaluate the number of roots of ##f(x;p)=0##. If no root is found then let ##b = p## and reiterate. If a root is found then let ##a = p## and reiterate.

The issue is, every root finder I see doesn't use a good value when no root is found. It returns an error message with an arbitrary value. I need a way to distinguish when a root is found or not...any ideas?

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fresh_42
Mentor
How about setting ##f(x;p)=p(x-x_0)^2## and examination of the consequences either by comparison of the terms or differentiation? Not sure how far you get with this approach, but it's an idea.

Math_QED
Homework Helper
2019 Award
Could you define double root please?

• joshmccraney
Gold Member
Could you define double root please?
Sorry, I can see I've been somewhat inaccurate. A double root implies the multiplicity of a root is 2. If a double root occurs at ##x=a## for a given ##p##, then ##f(a;p) = f'(a;p) = 0##. This is a necessary condition...

So I just solved it! Following your question I applied the above two equation equalities and out came a good answer!

Last edited:
• Math_QED