JEE Advanced · Mathematics · 25. AOD

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} & \text { if } x \neq 0 \\ \frac{7}{3} & \text { if } x=0\end{cases}\)
Then which of the following statements is (are) TRUE?

A The point \(x=0\) is a point of local maxima of \(f\)
B The point \(x=0\) is a point of local minima of \(f\)
C Number of points of local maxima of \(f\) in the interval \([\pi, 6 \pi]\) is 3
D Number of points of local minima of \(f\) in the interval \([2 \pi, 4 \pi]\) is 1

Verified Solution

Answer & Solution

Correct Answer

(B) The point \(x=0\) is a point of local minima of \(f\)

Step-by-step Solution

Detailed explanation

since \(3>\frac{7}{3} \Rightarrow 3>f(0)\)
\(\Rightarrow x=0\) is local minima ....option (B) is correct
Now,
\(\begin{aligned}
& f(x)=\frac{6 x+\sin x}{2 x+\sin x}=1+\frac{4 x}{2 x+\sin x} \\
& f^{\prime}(x)=\frac{4[(2 x+\sin x) \cdot 1-x(2+\cos x)]}{(2 x+\sin x)^2}=\frac{4(\sin x-x \cos x)}{(2 x+\sin x)^2} \\
& =\frac{4 \cos x(\tan x-x)}{(2 x+\sin x)^2}
\end{aligned}\)

Options (C) \& (D) are also correct.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Column I	Column II
(A) In \(R ^2\), if the magnitude of the projection vector of the vector \(\alpha \hat{ i }+\beta \hat{ j }\) on \(\sqrt{3} \hat{ i }+\hat{ j }\) is \(\sqrt{3}\) and if \(\alpha=2+\sqrt{3} \beta\), then possible value (s) of \(\|\alpha\|\) is (are)	(P) 1
(B) Let a and b be real numbers such that the function \(f ( x )=\left\{\begin{array}{cc}-3 ax ^2-2, & x <1 \\ bx + a ^2, & x \geq 1\end{array}\right.\) is differentiable for all \(x \in R\). Then possible value ( s ) of a is (are)	(Q) 2
(C)Let \(\omega \neq 1\) be a complex cube roots of unity. If \(\left(3-3 \omega+2 \omega^2\right)^{4 n+3}\) \(+~\left(2+3 \omega-3 \omega^2\right)^{4 n+3}\) \(~+\left(-3+2 \omega+3 \omega^2\right)^{4 n+3}=0\) then possible value \((s)\) of \(n\) is (are)	(R) 3
(D) Let the harmonic mean of two positive real numbers \(a\) and \(b 4\). If \(q\) is a positive real number such that \(a , 5, q , b\) is an arithmetic progression, then the value \(( s )\) of \(\| q - a \|\) is (are)	(S) 4
	(T) 5

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Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by\(f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} & \text { if } x \neq 0 \\ \frac{7}{3} & \text { if } x=0\end{cases}\)Then which of the following statements is (are) TRUE?

Step-by-step Solution

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by
\(f(x)= \begin{cases}\frac{6 x+\sin x}{2 x+\sin x} & \text { if } x \neq 0 \\ \frac{7}{3} & \text { if } x=0\end{cases}\)
Then which of the following statements is (are) TRUE?