Let fxbe a polynomial of degree $$5$$ such that $$x=$$±$$1are$$ its critical points. If limx→$$02+fxx3=4$$, then which one of the following is not true?

Question

Let  fxbe a polynomial of degree $$5$$ such that  $$x=$$±$$1are$$ its critical points. If  limx→$$02+fxx3=4$$, then which one of the following is not true?

Infinity Learn · Accepted Answer

Let $$fx=ax5+bx4+cx3$$.  Thenlimx→$$02+ax5+bx4+cx3x3=4$$⇒$$2+c=4$$⇒$$c=2Also$$ f'$$x=5ax4+4bx3+6x2=x25ax2+4bx+6$$.Since $$x=$$±$$1$$ are points of extremum, we havef'$$1=0$$ and f'−$$1=0$$⇒$$5a+4b+6=0$$ and $$5a$$−$$4b+6=0$$⇒$$b=0$$,$$a=$$−$$65$$⇒$$fx=$$−$$65x5+2x3$$⇒ f'$$x=$$−$$6x4+6x2$$ ⇒f''$$x=-24x3+12x$$ Now f''$$-1=12$$>$$0$$ and f''$$1=-12$$<$$0$$⇒minimum at x $$=$$ $$-1$$ and maximum at x $$=$$ $$1$$

Let $f (x)$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim_{x \to 0} (2 + \frac{f (x)}{x^{3}}) = 4$ , then which one of the following is not true?

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Let fxbe a polynomial of degree 5 such that x=±1are its critical points. If limx→02+fxx3=4, then which one of the following is not true?

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Let $f (x)$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim_{x \to 0} (2 + \frac{f (x)}{x^{3}}) = 4$ , then which one of the following is not true?