SOLUTION: Find the equations on the real number p for which {{{x^4 -4p^3x+12>0}}} for all real numbers x .
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Question 1037050: Find the equations on the real number p for which for all real numbers x .
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Find the extreme points of the function :
==> f'(x) = = 0
==>
==> There is only one critical value of x = p. ( The function has no real roots in terms of p and so there no critical values coming from this function.)
Because f"(p) = >0, the second derivative test tells that there is a relative (in fact, absolute) minimum at x = p.
We want , or
.
Since always, it follows that , or .
The solution set to the last inequality is .
Therefore as long as p is in the open interval (,), for all real numbers x .
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