AP EAMCET · Maths · Application of Derivatives
If the function \(f(x)=x^3+b x^2+c x-6\) satisfies all the conditions of Rolle's theorem in \([1,3]\) and \(f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0\), then \(b c=\)
- A \(18\)
- B \(-66\)
- C \(38\)
- D \(-46\)
Answer & Solution
Correct Answer
(B) \(-66\)
Step-by-step Solution
Detailed explanation
\(f(1)=f(3)\) \(1^3+b(1)^2+c(1)-6 = 3^3+b(3)^2+c(3)-6\) \(1+b+c-6 = 27+9b+3c-6\) \(b+c-5 = 9b+3c+21\) \(8b+2c = -26 \implies 4b+c=-13 \quad (1)\) \(f'(x) = 3x^2+2bx+c\) Let \(x_0 = \frac{2\sqrt{3}+1}{\sqrt{3}} = 2+\frac{1}{\sqrt{3}}\).…
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