Question

The condition $f(x)={{x}^{3}}+p{{x}^{2}}+qx+r$ $\left( x\in R \right)$ to have no extreme value is :
(1) ${{p}^{2}}<3q$
(2) $2{{p}^{2}}< q$
(3) ${{p}^{2}}< \dfrac{1}{4q}$
(4) ${{p}^{2}}>3q$

Answer 1

Here in this question, we have been asked to find the condition for which the given expression $f(x)={{x}^{3}}+p{{x}^{2}}+qx+r$ has no extreme values. From the basic concepts, we know that the extreme values of an expression $f\left( x \right)$ occur when $f'\left( x \right)=0$ . By using this we will evaluate the condition.

Complete step by step solution:
Now considering from the question, we have been asked to find the condition for which the given expression $f(x)={{x}^{3}}+p{{x}^{2}}+qx+r$ has no extreme values.
From the basic concepts, we know that the extreme values of an expression $f\left( x \right)$ occur when $f'\left( x \right)=0$ . By using this we will evaluate the condition.
Now by using $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$ formula from the concept of derivations we can find the first derivative of $f\left( x \right)$ which will be given as follows: $f'\left( x \right)=3{{x}^{2}}+2px+q$ .
As there exists no extreme values $f'\left( x \right)\ne 0$ this implies that $f'\left( x \right)=3{{x}^{2}}+2px+q\ne 0$ .
Hence we can say that the quadratic expression $f'\left( x \right)$ should not have any real roots that mean that the discriminant of the expression should be less than zero.
Since from the basic concepts of quadratic expressions, we know that the quadratic expression $a{{x}^{2}}+bx+c$ has real roots only when the discriminant ${{b}^{2}}-4ac$ of the given expression is greater than or equal to zero.
The discriminant of the expression will be given as $4{{p}^{2}}-12q$ .
Here our required condition is
$\begin{aligned} & 4{{p}^{2}}-12q<0 \\\ & \Rightarrow {{p}^{2}}-3q<0 \\\ & \Rightarrow {{p}^{2}}<3q \\\ \end{aligned}$ .
Therefore we can conclude that the condition for which the given expression $f(x)={{x}^{3}}+p{{x}^{2}}+qx+r$ has no extreme values will be given as ${{p}^{2}}<3q$ .
Hence we will mark the option “1” as correct.

Note: While answering questions of this type, we should be sure with the concepts that we are going to apply during the process of answering questions of this type. The extreme values for any expression $f\left( x \right)$ will be occurred at $f'\left( x \right)=0$ and the maximum value will occur when $f''\left( x \right)<0$ and the minimum value will occur at $f''\left( x \right)>0$.