The condition that f(x)=a x^3+b x^2+c x+d has | vedclass

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(D) Given the function $f(x) = a x^3 + b x^2 + c x + d$.To find the extreme values, we differentiate $f(x)$ with respect to $x$:$f'(x) = 3 a x^2 + 2 b

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$b^2 < 3 a c$

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(D) Given the function $f(x) = a x^3 + b x^2 + c x + d$.To find the extreme values, we differentiate $f(x)$ with respect to $x$:$f'(x) = 3 a x^2 + 2 b x + c$.For the function to have no extreme values, the derivative $f'(x)$ must not change its sign, which means the quadratic equation $3 a x^2 + 2 b x + c = 0$ must have no real roots or a repeated root such that the sign does not change.This occurs when the discriminant $D$ of the quadratic equation $3 a x^2 + 2 b x + c = 0$ is less than or equal to $0$.The discriminant $D = (2 b)^2 - 4(3 a)(c) = 4 b^2 - 12 a c$.Setting $D $4 b^2 - 12 a c $b^2 - 3 a c $b^2 Thus, the condition for no extreme value is $b^2 < 3 a c$.

The condition that $f(x)=a x^3+b x^2+c x+d$ has no extreme value, is

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