TS EAMCET · Maths · Application of Derivatives
The condition \(f(x)=x^3+p x^2+q x+r(x \in R)\) to have no extreme value, is
- A \(p^2 < 3 q\)
- B \(2 p^2 < q\)
- C \(p^2 < \frac{1}{4} q\)
- D \(p^2>3 q\)
Answer & Solution
Correct Answer
(A) \(p^2 < 3 q\)
Step-by-step Solution
Detailed explanation
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, \(\quad b^2-4 a c < 0\) \(\Rightarrow \quad 4 p^2-4 \times 3 \times q < 0\) \(\begin{array}{rrr}\Rightarrow & 4 p^2-12 q < 0 \\ \Rightarrow & p^2 < 3 q\end{array}\)
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