SOLUTION: Find the value of a for which {{{ f(x)=-2x^3+3(a+2)x^2-12ax+4a^2 }}} is tangent to the positive x-axis and has a relative maximumn at that point of contact.

Question 1036323: Find the value of a for which is tangent to the positive x-axis and has a relative maximumn at that point of contact.

Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Let (b,0) be the point of tangency.
Now the derivative is given by f'(x) = .
Setting this to 0 to find the critical values, we get .
==> x = 2 or x = a.
Case(i). Let b = a.
==>
==> , after reduction.
==> a = 0 (double root), or a = 2.
Discard a = 0, because if it is to be the x-coordinate of the point of tangency, a has to be positive. (Remember tangency to positive x-axis.)
==> a = 2.
==> .
But this function has to be discarded as well, because even though f'(2) = 0, f"(2) = 0, and hence there is no maximum at that point, but a point of inflection.

Case (ii). Let b = 2.
==>
==>
==> ==> (a-1)(a-2) = 0 ==> a = 1 or a = 2.
Now we already know what happens when a = 2, and so we proceed letting a = 1.
==>
By using the 2nd derivative test, we find that there is a relative maximum
at x = 2. (There is relative min at x = 1.)
Therefore .

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