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Differentiated function (Posted on 2019-06-22) Difficulty: 3 of 5
Let f:ℝ→ℝ be twice differentiable such that f(0)=2, f'(0)=-2, f(1)=1

Find for at-least how many c ∈ (0, 1)

f(c).f'(c)+f''(c)=0

is satisfied.

No Solution Yet Submitted by Danish Ahmed Khan    
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Either solution or confused Comment 1 of 1
The simplest function that fits seems to be the quadratic f(x)=x^2-2x+2.

But then we require (c^2-2c+2)(2c-2)+2=0 which is a cubic with solutions around {-.9, 1.3, 2.4}.  There are none on the interval (0,1).  This implies the there are at-least zero c.

When I bumped up to a cubic there is more wiggle room.  f(x)=ax^3+(1-a)x^2-2x+2 fits the qualifications for any a.

Playing with Desmos, I can find values of a with 2 solutions (if a=-0.5 then c=0.206 or c=0.791) or 1 solution (if a=0.5 then c=0.347) but I cannot find where there are zero values of c.
 
A question on notation then: Does  f:ℝ→ℝ rule out quadratic equations?  

  Posted by Jer on 2019-06-25 09:17:38
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