Home
Class 12
MATHS
Let f:(0,oo)->R be a differentiable func...

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

A

`underset(xrarr0^(+))limf'((1)/(x))=1`

B

`underset(xrarr0^(+))limxf((1)/(x))=2`

C

`underset(xrarr0^(+))limx^(2)f'(x)=0`

D

`|f(x)|le2" for all "x in (0,2)`

Text Solution

Verified by Experts

`f'(x)+(f(x))/(x)=2`
`rArr" "xf'(x)+f(x)=2x`
`rArr" "int d(x.f(x))=int 2xdx`
`rArr" "xf(x)=x+(c)/(x)" "(c ne 0 as f(1) ne1)`
`underset(xrarr0^(+))limf'((1)/(x))=underset(xrarr0^(+))lim(1-cx^(2))=1`
`underset(xrarr0^(+))limxf((1)/(x))=underset(xrarr0^(+))lim(1+cx^(2))=1`
`underset(xrarr0^(+))limx^(2)f'(x)=underset(xrarr0^(+))lim(x^(2)-c)=-c ne 0`
`underset(xrarr0^(+))limf(x)=oo or -oo" so option (4) is incorrect "`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIATION

    CENGAGE|Exercise Matrix Match Type|1 Videos

  • DIFFERENTIATION

    CENGAGE|Exercise Numerical Value Type|3 Videos

  • DIFFERENTIATION

    CENGAGE|Exercise JEE Previous Year|15 Videos

  • DIFFERENTIAL EQUATIONS

    CENGAGE|Exercise Question Bank|7 Videos

  • DOT PRODUCT

    CENGAGE|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

If f:RR-> RR is a differentiable function such that f(x) > 2f(x) for all x in RR and f(0)=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 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->R be a twice differentiable function such that f(x+pi)=f(x) and f''(x)+f(x)geq0 for all x in Rdot Show that f(x)geq0 for all x in Rdot

Let f(x) be a differentiable function such that f(x)=x^2 +int_0^x e^-t f(x-t) dt then int_0^1 f(x) dx=

Let f:[1,oo] be a differentiable function such that f(1)=2. If 6int_1^xf(t)dt=3xf(x)-x^3 for all xgeq1, then the value of f(2) is

Let f: RvecR be a continuous and differentiable function such that (f(x^2+1))^(sqrt(x))=5forAAx in (0,oo), then the value of (f((16+y^2)/(y^2)))^(4/(sqrt(y)))fore a c hy in (0,oo) is equal to (a) 5 (b) 25 (c) 125 (d) 625

Let f be differentiable function such that f'(x)=7-3/4(f(x))/x,(xgt0) and f(1)ne4" Then " lim_(xto0^+) xf(1/x)

Let f:[0,oo)rarrR be a continuous function such that f(x)=1-2x+int_(0)^(x)e^(x-t)f(t)dt" for all "x in [0, oo). Then, which of the following statements(s) is (are)) TRUE?

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,