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Calculus (Graphing Polynomial Curves)?
The question is "a quartic polynomial function f is defined by f(x) = x^4 + kx^2 + 8x +1 where k is some constant. The function f has a horizontal tangent and a point of inflection for some value of x. Determine the value of x for which this is true, then determine what must be the value of k"
I have no clue how to do this one. Any help would be greatly appreciated. Thanks.
2 Answers
- falzoonLv 78 years agoFavorite Answer
f(x) = x^4 + kx^2 + 8x + 1
Horizontal tangent occurs when f '(x) = 0.
f '(x) = 4x^3 + 2kx + 8 = 0
Dividing by 2 gives : 2x^3 + kx + 4 = 0 ..... (1)
Point of inflection occurs when f "(x) = 0.
f "(x) = 12x^2 + 2k = 0
Therefore, x = ± √(-k/6)
Plug both of these results back into (1) :
(a) 2[-√(-k/6)]^3 + k[-√(-k/6)] + 4 = 0
implies k = 6(-1)^(1/3) or k = -6(-1)^(2/3),
which are both complex solutions so we'll ignore those.
(b) 2[+√(-k/6)]^3 + k[+√(-k/6)] + 4 = 0
implies k = -6.
Thus, the polynomial is f(x) = x^4 - 6x^2 + 8x + 1
