For every twice differentiable function f:R→-2, 2 with f02+f′02=85 , which of the following statement(s) is (are) TRUE?

(A) There exist r,s∈R, where r<s, such that f is one-one on the open interval (r,s)

(B) There exists x0∈-4,0 such that f′x0≤1

(D) There exists α∈-4, 4 such that fα+f′′α=0 and f′α ≠0

JEE Advanced · Mathematics · 25. AOD

For every twice differentiable function $f : R \to [- 2, 2] w i t h {(f (0))}^{2} + {(f^{'} (0))}^{2} = 85$ , which of the following statement(s) is (are) TRUE?

A There exist $r, s \in R, w h e r e r < s,$ such that $f$ is one-one on the open interval $(r, s)$
B There exists $x_{0} \in (- 4,0)$ such that $|f^{'} (x_{0})| \leq 1$
C $\lim_{x \to \infty} f (x) = 1$
D There exists $α \in (- 4, 4)$ such that $f (α) + f^{''} (α) = 0 a n d f^{'} (α) \neq 0$

Verified Solution

Answer & Solution

Correct Answer

(A) There exist $r, s \in R, w h e r e r < s,$ such that $f$ is one-one on the open interval $(r, s)$

Step-by-step Solution

Detailed explanation

f (x)

can't be constant throughout the domain. Hence we can find

x \in (r, s)

such that

ƒ (x)

is one-one option

(A)

is true.

(B)

|f^{'} (x)| = |\frac{f (0) - f (- 4)}{4}| \leq 1 (L M V T)

(C)

f (x) = \sin (\sqrt{85 x})

satisfies given condition
But

\lim_{x \to \infty} \sin (\sqrt{85}) D . N . E

\Rightarrow

Incorrect

(D)

g (x) = f^{2} (x) + {(f^{'} (x))}^{2}

|f^{'} (x_{1})| \leq 1

(by LMVT)

|f (x_{1})| \leq 2

(given)

\Rightarrow g (x_{1}) \leq 5 \exists x_{1} \in (- 4,0)

Similarly

g (x_{2}) \leq 5 \exists x_{2} \in (0,4)

g (0) = 85 \Rightarrow g (x)

has maxima in

(x_{1}, x_{2}) s a y a t α

g^{'} (α) = 0 a n d g (α) \geq 85

2 f^{'} (α) (f (α) + f^{''} (α)) = 0

f^{'} (α) = 0 \Rightarrow g (α) = f^{2} (α) \geq 85

not possible

\Rightarrow f (α) + f^{''} (α) = 0 \exists α \in (x_{1}, x_{2}) \in (- 4,4)

(D)

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.

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Tangents are drawn from the point $P(3,4)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points $A$ and $B$.Question:
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JEE Advanced 2010 Medium

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of $n$ for which the function $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$,is	(1)	8
(Q)	The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}$,is an increasing function on $\mathbb{R}$,is	(2)	9
(R)	The smallest natural number $n$ which is greater than 5,such that $x=3$ is a point of local minima of $h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)$,is	(3)	5
(S)	Number of $x_0 \in \mathbb{R}$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}$,is NOT differentiable at $x_0$,is	(4)	6
		(5)	10

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Column I	Column II
A. Carbonate	P. Siderite
B. Sulphide	Q. Malachite
C. Hydroxide	R. Bauxite
D. Oxide	S. Calamine
	T. Argentite

	LIST - I		LIST - II
(P)	The minimum value of \(n\) for which the function \(f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]\) is continuous on the interval \([1,2]\),is	(1)	8
(Q)	The minimum value of \(n\) for which \(g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}\),is an increasing function on \(\mathbb{R}\),is	(2)	9
(R)	The smallest natural number \(n\) which is greater than 5,such that \(x=3\) is a point of local minima of \(h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)\),is	(3)	5
(S)	Number of \(x_0 \in \mathbb{R}\) such that \(l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}\),is NOT differentiable at \(x_0\),is	(4)	6
		(5)	10