AP EAMCET · Maths · Application of Derivatives
The maximum value of ' \(a\) ' such that the second derivative of \(x^4+a x^3+\frac{3 x^2}{2}+1\) is positive for all real \(x\) is
- A \(3\)
- B \(-3\)
- C \(2\)
- D \(-2\)
Answer & Solution
Correct Answer
(C) \(2\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { } \because f(x)=x^4+a x^3+\frac{3 x^2}{2}+1 \\ & \Rightarrow f^{\prime \prime}(x)=12 x^2+6 a x+3 \\ & \because f^{\prime \prime}(x)>0 \Rightarrow 4 x^2+3 a x+1>0 \\ & \Rightarrow\left(2 x+\frac{1}{2} a\right)^2+\left(1-\frac{a^2}{4}\right)>0 \\ &…
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