JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f\) and \(g\) be twice differentiable functions on \(R\) such that \(f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x\) \(f^{\prime}(1)=4 g^{\prime}(1)-3=9\) \(f(2)=3 g(2)=12\) Then which of the following is NOT true ?
- A \(g(-2)-f(-2)=20\)
- B \(9\)
- C \(\left|f^{\prime}(x)-g^{\prime}(x)\right| < 6 \Rightarrow-1 < x < 1\)
- D There exists \(x _0 \in\left(1, \frac{3}{2}\right)\) such that \(f \left( x _0\right)= g \left( x _0\right)\)
Answer & Solution
Correct Answer
(B) \(9\)
Step-by-step Solution
Detailed explanation
\(f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x\) \(f^{\prime}(1)=4 g^{\prime}(1)-3=9\) \(f(2)=3 g(2)=12\) By integrating \((1)\) \(f^{\prime}(x)=g^{\prime}(x)+6 \frac{x^2}{2}+C\) At \(x=1\), \(f^{\prime}(1)=g^{\prime}(1)+3+C\) \(\Rightarrow 9=4+3+C \Rightarrow C=3\)…
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