Let f: R to R,f(x) = ax^3 + bx^2 + cx + d hav | vedclass

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(C) For a cubic function $f(x) = ax^3 + bx^2 + cx + d$ to have no extreme values,its derivative $f'(x) = 3ax^2 + 2bx + c$ must not change sign.This me

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$(3a + 2b + c)c \ge 0$

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(C) For a cubic function $f(x) = ax^3 + bx^2 + cx + d$ to have no extreme values,its derivative $f'(x) = 3ax^2 + 2bx + c$ must not change sign.This means $f'(x)$ must be either always $\ge 0$ or always $\le 0$ for all $x \in R$.This implies that the quadratic expression $g(x) = 3ax^2 + 2bx + c$ does not change sign.For a quadratic $g(x) = Ax^2 + Bx + C$ to maintain a constant sign,its discriminant $D = B^2 - 4AC$ must be $\le 0$.Here,$D = (2b)^2 - 4(3a)(c) = 4b^2 - 12ac \le 0$,which implies $b^2 \le 3ac$.If $f'(x)$ does not change sign,then $f'(x_1)$ and $f'(x_2)$ must have the same sign for any $x_1, x_2 \in R$.Specifically,$f'(1) = 3a + 2b + c$ and $f'(0) = c$ must have the same sign or be zero.Therefore,their product must be non-negative: $(3a + 2b + c)c \ge 0$.

Let $f: R \to R$,$f(x) = ax^3 + bx^2 + cx + d$ have no extreme value. Then which of the following is always correct?

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