JEE Mains · Maths · STD 12 - 1. relation and function
Consider the function \(\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathrm{R}\) defined by \(f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1\). Consider the statements \((I)\) The curve \(y=f(x)\) intersects the \(x\)-axis exactly at one point \((II)\) The curve \(y=f(x)\) intersects the \(x\)-axis at \(\mathrm{x}=\cos \frac{\pi}{12}\) Then
- A Only \((II)\) is correct
- B Both \((I)\) and \((II)\) are incorrect
- C Only\( (I)\) is correct
- D Both \((I)\) and \((II)\) are correct
Answer & Solution
Correct Answer
(D) Both \((I)\) and \((II)\) are correct
Step-by-step Solution
Detailed explanation
\(\mathrm{f}^{\prime}(\mathrm{x})=12 \sqrt{2} \mathrm{x}^2-3 \sqrt{2} \geq 0 \text { for }\left[\frac{1}{2}, 1\right]\) \(\mathrm{f}\left(\frac{1}{2}\right)<0\) \(\mathrm{f}(1)>0 \Rightarrow(\mathrm{A})\) is correct. \(f(x)=\sqrt{2}\left(4 x^3-3 x\right)-1=0\) Let…
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