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Let f(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If lim(x→0) (2 + f(x)/x^3) = 4, then which one of the following is not true? (1) f is an odd function (2) f(1) - 4f(-1) = 4 (3) x = 1 is a point of minima and x = -1 is a point of maxima of f (4) x = 1 is a point of maxima and x = -1 is a point of minima of f
- (1) f is an odd function
- (2) f(1) - 4f(-1) = 4
- (3) x = 1 is a point of minima and x = -1 is a point of maxima of f
- (4) x = 1 is a point of maxima and x = -1 is a point of minima of f
Correct answer: (4) x = 1 is a point of maxima and x = -1 is a point of minima of f
Solution
The correct option is not true because the critical points at x = ±1 indicate that one is a maximum and the other a minimum, but the specific nature of these points depends on the behavior of the polynomial around them. Given that x = ±1 are critical points and the polynomial is of odd degree, it is more consistent with the properties of polynomials that one would be a maximum and the other a minimum, making the assertion in option (4) incorrect.
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