Let f(x) = x^{12} - x^9 + x^4 - x + 1. Which | vedclass

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(D) Given $f(x) = x^{12} - x^9 + x^4 - x + 1$.First,check if $f$ is one-one: $f(0) = 1$ and $f(1) = 1 - 1 + 1 - 1 + 1 = 1$. Since $f(0) = f(1)$,$f$ is

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$f$ takes only positive values

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(D) Given $f(x) = x^{12} - x^9 + x^4 - x + 1$.First,check if $f$ is one-one: $f(0) = 1$ and $f(1) = 1 - 1 + 1 - 1 + 1 = 1$. Since $f(0) = f(1)$,$f$ is not one-one.Next,check if $f$ takes only positive values:Case $1$: If $x \leq 0$,then $f(x) = x^{12} - x^9 + x^4 - x + 1$. Since $x^{12} \geq 0$,$-x^9 \geq 0$,$x^4 \geq 0$,and $-x \geq 0$,we have $f(x) \geq 1 > 0$.Case $2$: If $x > 1$,then $f(x) = x^9(x^3 - 1) + x(x^3 - 1) + 1$. Since $x > 1$,$x^3 - 1 > 0$,so $f(x) > 1 > 0$.Case $3$: If $0  0$,$1 - x^5 > 0$,and $x^{12} > 0$,so $f(x) > 0$.Since $f(x) > 0$ for all $x \in \mathbb{R}$,$f$ has no real roots. Thus,option $(d)$ is correct.

Let $f(x) = x^{12} - x^9 + x^4 - x + 1$. Which of the following is true?

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