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

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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 $x$: $f'(x) = 3 a x^2 + 2 b x + c$. $A$ function has no extreme value if its derivative $f'(x)$ does not change sign,which occurs when the quadratic equation $f'(x) = 0$ has no real roots or has equal roots such that the sign does not change. For the quadratic equation $3 a x^2 + 2 b x + c = 0$ to have no real roots,its discriminant $D$ must be less than $0$. $D = (2 b)^2 - 4(3 a)(c) $4 b^2 - 12 a c Dividing by $4$,we get $b^2 - 3 a c 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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