Let f(x) and g(x) be twice differentiable function in R and satisfying the equation f prime prime(x)=g prime prime(x) such that f prime(1)=2 g prime(1)=4 and f(2)=3 g(2)=9, then [where operatornamesgn x dentoes the signum function of x.]

Question

Let f(x) and g(x) be twice differentiable function in R and satisfying the equation f prime prime(x)=g prime prime(x) such that f prime(1)=2 g prime(1)=4 and f(2)=3 g(2)=9, then [where operatornamesgn x dentoes the signum function of x.] (A) f(x)+g(x)=3 x+5 (B) if |f(x)-g(x)|<1 then sgn x=-1 (C) f(3)-g(3)=8 (D) area bounded between the curve y=f(x) and y=g(x) in [0,2] is 8 .

Tardigrade Admin · Accepted Answer

Given f prime prime(x)=g prime prime(x) f prime( x )= g prime( x )+λ....(i) text and f(x)=g(x)+λ x+μ....(ii) text According to question f prime(1)=4, g prime(1)=2....(iii) text and f(2)=9 text and g(2)=3 ....(iv) ∵ text From (i), (ii), (iii) and (iv) λ=4=2 ∴ f ( x )= g ( x )+2 x +2

Q. Let $f (x)$ and $g (x)$ be twice differentiable function in R and satisfying the equation $f^{''} (x) = g^{''} (x)$ such that $f^{'} (1) = 2 g^{'} (1) = 4$ and $f (2) = 3 g (2) = 9$ , then
[where $sgn x$ dentoes the signum function of $x$ .]

$f (x) + g (x) = 3 x + 5$

if $∣ f (x) - g (x) ∣ < 1$ then sgn $x = - 1$

$f (3) - g (3) = 8$

area bounded between the curve $y = f (x)$ and $y = g (x)$ in $[0, 2]$ is 8 .

Given $f^{''} (x) = g^{''} (x)$
$f^{'} (x) = g^{'} (x) + λ$ ....(i)
$and f (x) = g (x) + λ x + μ$ ....(ii)
$According to question$
$f^{'} (1) = 4, g^{'} (1) = 2$ ....(iii)
$and f (2) = 9 and g (2) = 3$ ....(iv)
$∵ From (i), (ii), (iii) and (iv)$
$λ = 4 = 2$
$∴ f (x) = g (x) + 2 x + 2$

Q. Let f(x) and g(x) be twice differentiable function in R and satisfying the equation f′′(x)=g′′(x) such that f′(1)=2g′(1)=4 and f(2)=3g(2)=9, then [where sgnx dentoes the signum function of x.]

f(x)+g(x)=3x+5

if ∣f(x)−g(x)∣<1 then sgn x=−1

f(3)−g(3)=8

area bounded between the curve y=f(x) and y=g(x) in [0,2] is 8 .