Suppose f(x) is a differentiable function g(x) is twice differentiable function such that f'(x)=g(x) & |f(x)|≤1 ∀x∈[−3,3] . If further (f(0))2+(g(0))2=9. Then which of the following is/are true?

There exists some c, c ∈ ( − 3 , 3 ) such that | f ' ( c ) | ≤ 2 3 , There exists some d , d ∈ ( − 3 , 3 ) such that f ( d ) + g ' ( d ) = 0 and g d ≠ 0 , Let h ( x ) = ( f ( x ) ) 2 + ( g ( x ) ) 2 and h ( x ) attains its local maximum at x = λ , λ ∈ ( − 3 , 3 ) , There exists a value of μ , μ ∈ ( − 3 , 3 ) such that g ( μ ) g " ( μ ) < 0

Suppose f(x) is a differentiable function g(x) is twice differentiable function such that f'(x)=g(x) & |f(x)|≤1 ∀x∈[−3,3] . If further (f(0))2+(g(0))2=9. Then which of the following is/are true?

h ( x ) = ( f ( x ) ) 2 + ( g ( x ) ) 2 U sin g L M V T , f o r f ( x ) i n ( − 3 , 0 ) c 1 ∈ ( − 3 , 0 ) f ' ( c 1 ) = f ( 0 ) − f ( − 3 ) 3 S i n c e | f ( x ) | ≤ 1 ⇒ − 1 ≤ f ( 0 ) ≤ 1 & − 1 ≤ − f ( − 3 ) ≤ 1 ⇒ − 2 ≤ f ( 0 ) − f ( − 3 ) ≤ 2 − 2 3 ≤ f ( 0 ) − f ( − 3 ) 3 ≤ 2 3 ⇒ | f ' ( c 1 ) | ≤ 2 3 N o w , t h e r e e x i s t s a t l e a s t o n e c ∈ ( c 1 , c 2 ) s u c h t h a t h ( x ) w i l l a t t a i n i t s m a x i m u m a t c ⇒ h ' ( c ) = 0 & h " ( c ) < 0 h ( c 1 ) = ( f ( c 1 ) ) 2 + ( g ( c 1 ) ) 2 ≤ 1 + 4 9 ≤ 13 9 h ' ( x ) = 2 f ( x ) f ' ( x ) + 2 g ( x ) g ' ( x ) = 2 f ( x ) g ( x ) + 2 g ( x ) g ' ( x ) , ( ∵ f ' ( x ) = g ( x ) ) 2 g ( x ) ( f ( x ) + g ' ( x ) ) h ' ( c ) = 0 ⇒ 2 g ( c ) ( f ( c ) + g ' ( c ) ) = 0 ⇒ f ( c ) = − g ' ( c ) h " ( x ) = 2 [ f ' ( x ) 2 + f ( x ) f " ( x ) + ( g ' ( x ) ) 2 + g ( x ) g " ( x ) ] = 2 [ ( g ( x ) ) 2 + f ( x ) g ' ( x ) + ( g ' ( x ) ) 2 + g ( x ) g " ( x ) ] , [ f ' ( x ) = g ( x ) h " ( c ) < 0 ⇒ ( g ( c ) ) 2 + f ( c ) g ' ( c ) + ( g ' ( c ) ) 2 + g ( c ) g ' ( c ) < 0 f ( c ) = − g ' ( c ) ( g ( c ) ) 2 − ( g ' ( c ) ) 2 + ( g ' ( c ) ) 2 + g ( c ) g " ( c ) < 0 g ( ( c ) ) 2 + g ( c ) g " ( c ) < 0 ⇒ g ( c ) g " ( c ) < 0. ∀ c ∈ ( − 3 , 3 )

There exists some c, c ∈ ( − 3 , 3 ) such that | f ' ( c ) | ≤ 2 3

There exists some d , d ∈ ( − 3 , 3 ) such that f ( d ) + g ' ( d ) = 0 and g d ≠ 0

There exists a value of μ , μ ∈ ( − 3 , 3 ) such that g ( μ ) g " ( μ ) < 0

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Suppose $f (x)$ is a differentiable function $g (x)$ is twice differentiable function such that $f' (x) = g (x)$ & $| f (x) | \leq 1 \forall x \in [- 3, 3]$ . If further ${(f (0))}^{2} + {(g (0))}^{2} = 9.$ Then which of the following is/are true?

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There exists some c, c $\in (- 3, 3)$ such that $| f' (c) | \leq \frac{2}{3}$

There exists some $d$ , $d \in (- 3, 3)$ such that $f (d) + g' (d) = 0$ and $g (d) \neq 0$

Let $h (x) = {(f (x))}^{2} + {(g (x))}^{2}$ and $h (x)$ attains its local maximum at $x = λ,$ $λ \in (- 3, 3)$

There exists a value of $μ$ , $μ \in (- 3, 3)$ such that $g (μ) g " (μ) < 0$

answer is A, B, C, D.

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Detailed Solution

$h (x) = {(f (x))}^{2} + {(g (x))}^{2} U \sin g L M V T, f o r f (x) i n (- 3, 0) c_{1} \in (- 3, 0) f' (c_{1}) = \frac{f (0) - f (- 3)}{3} S i n c e | f (x) | \leq 1 \Rightarrow - 1 \leq f (0) \leq 1 & - 1 \leq - f (- 3) \leq 1 \Rightarrow - 2 \leq f (0) - f (- 3) \leq 2 - \frac{2}{3} \leq \frac{f (0) - f (- 3)}{3} \leq \frac{2}{3} \Rightarrow | f' (c_{1}) | \leq \frac{2}{3} N o w, t h e r e e x i s t s a t l e a s t o n e c \in (c_{1}, c_{2}) s u c h t h a t h (x) w i l l a t t a i n i t s m a x i m u m a t c \Rightarrow h' (c) = 0 & h " (c) < 0 h (c_{1}) = {(f (c_{1}))}^{2} + {(g (c_{1}))}^{2} \leq 1 + \frac{4}{9} \leq \frac{13}{9} h' (x) = 2 f (x) f' (x) + 2 g (x) g' (x) = 2 f (x) g (x) + 2 g (x) g' (x), (∵ f' (x) = g (x)) 2 g (x) (f (x) + g' (x)) h' (c) = 0 \Rightarrow 2 g (c) (f (c) + g' (c)) = 0 \Rightarrow f (c) = - g' (c) h " (x) = 2 [f' {(x)}^{2} + f (x) f " (x) + {(g' (x))}^{2} + g (x) g " (x)] = 2 [{(g (x))}^{2} + f (x) g' (x) + {(g' (x))}^{2} + g (x) g " (x)], [f' (x) = g (x) h " (c) < 0 \Rightarrow {(g (c))}^{2} + f (c) g' (c) + {(g' (c))}^{2} + g (c) g' (c) < 0 f (c) = - g' (c) {(g (c))}^{2} - {(g' (c))}^{2} + {(g' (c))}^{2} + g (c) g " (c) < 0 g {((c))}^{2} + g (c) g " (c) < 0 \Rightarrow g (c) g " (c) < 0. \forall c \in (- 3, 3)$

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