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Calculus/Looking for relative minima, maxima and extrema

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Expert: Alon Mandes - 11/30/2009


Question
Graph the fourth-degree polynomial p(x)=x^4+ax^2+1 for various values of the constant a.
a)Determine the values of a for which p has exactly one relative minimum.
b)Determine the values of a for which p has exactly one relative maximum.
c)Determine the values of a for which p has exactly two relative minima.

d)Show that the graph of p cannot have exactly two relative extrema.

Answer
p(x)=x^4+ax^2+1
p'(x)=4x^3+2ax
p'(x)=0 : 2x^3+a=0 -- > xo=-(a/2)^(1/3) . We can see that we can have only 1 value
of x for extremum.
p''(x)=12x^2+2a
p''(xo)=12(a/2)^(2/3)+2a .
We get max when p''(xo)<0 , that means : -2a>12(a/2)^(2/3) --> a>56
We get min when p''(xo)>0 , that means : -2a<12(a/2)^(2/3) --> a<56

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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M.A in Mathematics & Bs.c in Electronics.

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