AP EAMCET · Maths · Application of Derivatives
The set of all real values of ' \(a\) ' such that the real valued function \(f(x)=x^3+2 a x^2+3(a+1) x+5\) is strictly increasing in its entire domain is
- A \(\left(-\infty,-\frac{3}{4}\right) \cup(3, \infty)\)
- B \(\left(-\frac{3}{4}, 3\right)\)
- C \((1,3)\)
- D \((-\infty, 1) \cup(3, \infty)\)
Answer & Solution
Correct Answer
(B) \(\left(-\frac{3}{4}, 3\right)\)
Step-by-step Solution
Detailed explanation
Given \(f(x)=x^3+2 a x^2+3(a+1) x+5\) Now, \(f^{\prime}(x)=3 x^2+4 a x+3(a+1)\) for strictly increasing of \(f(x)\) in its entire domain \(\begin{aligned} & (4 a)^2-4 \times 3 \times 3(a+1) \lt 0 \\ & \Rightarrow 16 a^2-36(a+1) \lt 0\end{aligned}\)…
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