Q107 / 160 · TS EAMCET 2007 · Mathematics

The condition $f(x)=x^3+p x^2+q x+r(x \in R)$ to have no extreme value, is

  • Correct answer
    A
    $p^2 < 3 q$
  • B
    $2 p^2 < q$
  • C
    $p^2 < \frac{1}{4} q$
  • D
    $p^2>3 q$

Solution

Given, $f(x)=x^3+p x^2+q x+r$ Now, $f^{\prime}(x)=3 x^2+2 x p+q$ Clearly $\quad f^{\prime}(x)>0$ Now, $\quad b^2-4 a c < 0$ $\Rightarrow \quad 4 p^2-4 \times 3 \times q < 0$ $\begin{array}{rrr}\Rightarrow & 4 p^2-12 q < 0 \\ \Rightarrow & p^2 < 3 q\end{array}$

Hence, the correct answer is option: 'A'.

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