Home
Class 12
MATHS
Let f : R rarr R and g : R rarr R be two...

Let `f : R rarr R and g : R rarr R` be two non-constant differentiable functions. If `f'(x) = (e^((f(x)-g(x))))g'(x) "for all x" in R and f(1) = g(2)=1`, then which of the following statement(s) is (are) TRUE?

A

`f(2) lt 1- log_(e)2`

B

`f(2) gt 1-log_(e)(2)`

C

`g(1) lt 1-log_(e)2`

D

`g(1) lt 1-log_(e)2`

Text Solution

Verified by Experts

The correct Answer is:
B, C
We have `f^(')(x)=(e^(f(x)-g(x))g^(')(x)),f(1)=g(2)=1`
`therefore e^(-f(x)f^(')(x) - e^(-g(x))g^(')(x)`
Integrating both sides, we get
`e^(-1)-e^(-f(2))=e^(-g(t)-e^(-1)`
`therefore e^(-f(2)) = 2e^(-1)-e^9-g(t) lt 2e^(-1)` `["as " e^(-g(t)) gt0]`
`rArr -f(2) lt log(2e^(-1))`
`rArr f(2) gt 1-log_(e)2 gt 1-log 2`
Similarly, `g(1) gt 1-log_(e)2`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise Single Correct Answer Type|37 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise Multiple Correct Answer Type|10 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise JEE Main Previous Year|12 Videos

  • DIFFERENT PRODUCTS OF VECTORS AND THEIR GEOMETRICAL APPLICATIONS

    CENGAGE|Exercise Exercise|337 Videos

  • DIFFERENTIATION

    CENGAGE|Exercise Numerical Value Type|3 Videos

Similar Questions

Explore conceptually related problems

Let f: R->R and g: R->R be two non-constant differentiable functions. If f^(prime)(x)=(e^((f(x)-g(x))))g^(prime)(x) for all x in R , and f(1)=g(2)=1 , then which of the following statement(s) is (are) TRUE? f(2) 1-(log)_e2 (c) g(1)>1-(log)_e2 (d) g(1)<1-(log)_e2

Let f:(0,oo)->R be a differentiable function such that f'(x)=2-f(x)/x for all x in (0,oo) and f(1)=1 , then

If f:R->R is a twice differentiable function such that f''(x) > 0 for all x in R, and f(1/2)=1/2. f(1)=1, then

Let f:R to R and h:R to R be differentiable functions such that f(x)=x^(3)+3x+2,g(f(x))=x and h(g(x))=x for all x in R . Then,

Let f be a differentiable function such that f(1) = 2 and f'(x) = f (x) for all x in R . If h(x)=f(f(x)), then h'(1) is equal to

Let f: R rarr R be a differentiable function satisfying f(x+y)=f(x)+f(y)+x^(2)y+xy^(2) for all real numbers x and y. If lim_(xrarr0) (f(x))/(x)=1, then The value of f(9) is

Let f: R rarr R be a differentiable function satisfying f(x+y)=f(x)+f(y)+x^(2)y+xy^(2) for all real numbers x and y. If lim_(xrarr0) (f(x))/(x)=1, then The value of f'(3) is

Let F : R to R be a thrice differentiable function . Suppose that F(1)=0,F(3)=-4 and F(x) lt 0 " for all" x in (1/2,3). f(x) = x F(x) for all x inR . The correct statement (s) is / are

Let f(x) and g(x) are two differentiable functions. If f(x)=g(x) , then show that f'(x)=g'(x) . Is the converse true ? Justify your answer

If for a function f : R to R f (x +y ) =F(x ) + f(y) for all x and y then f(0) is