AP EAMCET · Maths · Application of Derivatives
If the function \(f(x)=x^3+2 p x^2+27 x+16\) is strictly increasing for all \(x \in R\), then the range of \(p\) is
- A \(\left(-\infty, \frac{-9}{2}\right) \cup\left(\frac{9}{2}, \infty\right)\)
- B \((-\infty,-9) \cup(9, \infty)\)
- C \(\left(\frac{-9}{2}, \frac{9}{2}\right)\)
- D \((-9,9)\)
Answer & Solution
Correct Answer
(C) \(\left(\frac{-9}{2}, \frac{9}{2}\right)\)
Step-by-step Solution
Detailed explanation
\[ \begin{aligned} & f(x)=x^3+2 p x^2+27 x+16 \\ & f^{\prime}(x)=3 x^2+4 p x+27 \end{aligned} \] \(f(x)\) is strictly increasing function, so \(f^{\prime}(x)>0\) \[ \Rightarrow \quad 3 x^2+4 p x+27>0 \] Possible only when its \(D < 0\)…
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