The condition that f(x) = ax^3 + bx^2 + cx + | vedclass

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(D) Given the function $f(x) = ax^3 + bx^2 + cx + d$. To find the extreme values,we first find the derivative of $f(x)$ with respect to $x$: $f'(x) =

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$b^2 < 3ac$

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(D) Given the function $f(x) = ax^3 + bx^2 + cx + d$. To find the extreme values,we first find the derivative of $f(x)$ with respect to $x$: $f'(x) = 3ax^2 + 2bx + c$. $A$ function has no extreme values if its derivative $f'(x)$ does not change sign,which means $f'(x) = 0$ has no real roots or has equal roots such that the sign does not change. For the quadratic equation $3ax^2 + 2bx + c = 0$ to have no real roots,its discriminant $D$ must be less than $0$. $D = (2b)^2 - 4(3a)(c) $4b^2 - 12ac Dividing by $4$,we get $b^2 - 3ac < 0$,which implies $b^2 < 3ac$.

The condition that $f(x) = ax^3 + bx^2 + cx + d$ has no extreme value is

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