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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

$b^2 < 3 a c$

Vedclass · Answer

(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

Similar Questions

In the reaction $\underset{\text{(Vapour)}}{C_2H_5OH} \xrightarrow[300^{\circ} C]{Cu} X$. The molecular formula of $X$ is

$A$ plank with a small block on top of it is undergoing vertical $SHM$. Its period is $2 \, s$. The minimum amplitude at which the block will separate from the plank is:

The events $A$ and $B$ have probabilities $0.25$ and $0.50$,respectively. The probability that both $A$ and $B$ occur simultaneously is $0.14$. Then,the probability that neither $A$ nor $B$ occurs is:

The string of a pendulum of length $l$ is displaced through $90^o$ from the vertical and released. Then,the minimum strength of the string in order to withstand the tension as the pendulum passes through the mean position is

An aqueous solution of sodium carbonate is alkaline. This is because sodium carbonate is a salt of which of the following?

Vedclass Test Series

Exam Paper Generator

Online Exam Module