`f(x)` is a continuous and bijective function on `Rdot` If `AAt in R ,` then the area bounded by `y=f(x),x=a-t ,x=a ,` and the x-axis is equal to the

Question

`f(x)` is a continuous and bijective function on `Rdot` If `AAt in R ,` then the area bounded by `y=f(x),x=a-t ,x=a ,` and the x-axis is equal to the

`f(x)` is a continuous and bijective function on `Rdot` If `AAt in R ,` then the area bounded by `y=f(x),x=a-t ,x=a ,` and the x-axis is equal to the area bounded by `y=f(x),x=a+t ,x=a ,` and the x-axis. Then prove that `int_(-lambda)^lambdaf^(-1)(x)dx=2alambda(gi v e nt h a tf(a)=0)dot`

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

Related Questions

Find the work done in each case : (a) When one mole of ideal gas in 10 liter container aat 1 atm, is allowed to enter a vaccuated bulb of capacity 100

2 Answers 4 Views

If ` vec r` is a vector of magnitude 21 and has direction ratios `2,-3a n d6,` then find ` vec rdot`

2 Answers 5 Views

The equation of the plane which passes through the line of intersection of planes ` vec rdot vec n_1=, q_1, vec rdot vec n_2=q_2` and the is parallel

2 Answers 4 Views

Let `f(x)` and `phi(x)` are two continuous function on `R` satisfying `phi(x)=int_(a)^(x)f(t)dt, a!=0` and another continuous function `g(x)` satisfyi

2 Answers 4 Views

If `sin^(-1) : [-1, 1] rarr [(pi)/(2), (3pi)/(2)] and cos^(-1) : [-1, 1] rarr [0, pi]` be two bijective functions, respectively inverse of bijective f

2 Answers 4 Views

Find area bounded by `y = f^(-1)(x), x = 10, x = 4` and x-axis. Given that area bounded by `y = f(x) , x = 2, x = 6` and x -axis is `30` sq. units, wh

2 Answers 7 Views

Consider a line `l: 2x -sqrt(3)y = 0` and a parameterized `C :x = tant, y=1/cost(0<=t<pi/2)` If the area of the part bounded by `l,C` and the y-axis i

2 Answers 6 Views

The maximum area of the triangle whose sides `a ,b` and `5sintheta),` and `(5sintheta,-5costheta),` where `theta in Rdot` The locus of its orthocentre

2 Answers 5 Views

If the coefficients of `(r-5)^(t h)` and `(2r-1)^(t h)` terms in the expansion of `(1+x)^(34)` are equal, find `rdot`

2 Answers 5 Views

If `omega` is any complex number such that `z omega=|z|^(2)` and `|z-barz|+|omega+baromega|=4`, then as `omega` varies, then the area bounded by the l

2 Answers 7 Views

Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Given `|int_(a-t)^(a)f(x)dx|=|int_(a)^(a+t)f(x)dx|AAtepsilonR`
or `int_(a-t)^(a)f(x)dx=-int_(a)^(a+t)f(x)dx`
[since `f(a)=0` and `f(x)` is monotonic]
or `f(a-t)=-f(a-t)` (differentiating both side w.r.t `t`)
or `f(a+t)=-f(a-t)=x` (say)
or `t=f^(-1)(x)-a`..........2
and `t=a-f^(-1)(-x)` ............3
From equations 3 and 2 `(a-f^(-1)(x))+(a-f^(1)(-x))=0`
or `int_(-lamda)^(lamda) f^(-1)(x)dx=1/2 int_(-lamda)^(lamda)(f^(-1)(x)+f^(-1)(-x))dx=2alamda`