Limiting till the degree 5

Algebra Level 3

Let $f(x)$ be a polynomial of degree $5$ such that $x= \pm 1$ are its critical points.

If $\displaystyle \lim_{x \to 0}\left(2+\frac{f(x)}{{x}^{3}}\right)=4$ , then which of the following is not true?

f

is an odd function

x=1

is a point of minima and

x=-1

is a point of maxima of

f

f(1)-4f(-1)=4

x=1

is a point of maxima and

x=-1

is a point of minima of

f

1 solution

A Former Brilliant Member
Sep 20, 2020

@Kriti Kamal , @Aryan Sanghi , I guessed it, but I want to ask if there really exists a polynomial whose maxima is -1 and minima +1? If yes, how? Also the question states what are the critical points, so the answer has to be one of the 2 options, right?

Vinayak Srivastava - 8 months, 2 weeks ago

Here is such a polynomial

Just invert the signs of this question's polynomial, i.e. $y = 1.2x^5 - 2x^3$ :)

@Vinayak Srivastava

Aryan Sanghi - 8 months, 2 weeks ago

Ohk, thanks! I think I messed with concept, I understand where I am wrong. What about second question?

Vinayak Srivastava - 8 months, 2 weeks ago

Aryan has answered the first question.

Critical points are those points where either function is not differentiable or its
derivative is zero.However,it is not mentioned that function is differentiable.But,from options you can conclude that function is differentiable.

Limiting till the degree 5

1 solution

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