Let f(x) be a twice-differentiable funct...

Let `f(x)` be a twice-differentiable function and `f''(0)=2.` Then evaluate `underset(xto0)lim(2f(x)-3f(2x)+f(4x))/(x^(2)).`

A

3a

B

2a

C

5a

D

4a

Text Solution

Verified by Experts

The correct Answer is:

A

Using L' Hospital'e rule, we have
`underset(x to 0)lim (2f(x)-3f(2x)+f(4x))/(x^(2))" "((0)/(0)"form")`
`underset(x to 0)lim (2f'(x)-6f'(2x)+4f'(4x))/(2x)" "((0)/(0)"form")`
`underset(x to 0)lim (2f''(x)-12f''(2x)+16f''(4x))/(2)" "("Using L' Hospital's rule")`
`=(2f''(0)-12f''(0)+16f''(0))/(2)=3f''(0)=3a`

Promotional Banner

Promotional Banner

Topper's Solved these Questions

CONTINUITY AND DIFFERENTIABILITY
OBJECTIVE RD SHARMA ENGLISH|Exercise Section I - Solved Mcqs|143 Videos
CONTINUITY AND DIFFERENTIABILITY
OBJECTIVE RD SHARMA ENGLISH|Exercise Section II - Assertion Reason Type|13 Videos
COMPLEX NUMBERS
OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test|58 Videos
DEFINITE INTEGRALS
OBJECTIVE RD SHARMA ENGLISH|Exercise Chapter Test 2|56 Videos

Similar Questions

Explore conceptually related problems

Let f(x) be a twice-differentiable function and f''(0)=2. Then evaluate lim_(xto0) (2f(x)-3f(2x)+f(4x))/(x^(2)).

Let f(x) be a twice-differentiable function and f"(0)=2. The evaluate: ("lim")_(xvec0)(2f(x)-3f(2x)+f(4x))/(x^2)

If f(x) is a twice differentiable function such that f(0)=f(1)=f(2)=0 . Then

Let f (x) be twice differentialbe function such that f'' (x) gt 0 in [0,2]. Then :

Let f : R to R be a differentiable function satisfying f'(3) + f'(2) = 0 , Then underset(x to 0) lim ((1+f(3+x)-f(3))/(1+f(2-x)-f(2)))^(1/x) is equal to

Let f(x) be a differentiable function in the interval (0, 2) then the value of int_(0)^(2)f(x)dx

Let f''(x) be continuous at x = 0 and f"(0) = 4 then value of lim_(x->0)(2f(x)-3f(2x)+f(4x))/(x^2)

Let f (x) be a twice differentiable function defined on (-oo,oo) such that f (x) =f (2-x)and f '((1)/(2 )) =f' ((1)/(4))=0. Then int _(-1) ^(1) f'(1+ x ) x ^(2) e ^(x ^(2))dx is equal to :

if f(x) be a twice differentiable function such that f(x) =x^(2) " for " x=1,2,3, then

Let f be a twice differentiable function such that f''(x)=-f(x)" and "f'(x)=g(x)."If "h'(x)=[f(x)]^(2)+[g(x)]^(2), h(1)=8" and "h(0)=2," then "h(2)=

OBJECTIVE RD SHARMA ENGLISH-CONTINUITY AND DIFFERENTIABILITY-Exercise

Let f(x) be a twice-differentiable function and f''(0)=2. Then evaluat...
Text Solution
|
The function f(x) = (4-x^(2))/(4x-x^(3)) is discontinuous at
Text Solution
|
Let f(x)=|x| and g(x)=|x^3| , then (a).f(x) and g(x) both are continuo...
Text Solution
|
The function f(x)=sin^(-1)(cosx) is discontinuous at x=0 (b) continuou...
Text Solution
|
The set of points where the function f(x)=x|x| is differentiable is...
Text Solution
|
On the interval I = [-2, 2], if the function f(x) = {{:((x+1)e^(-((1)/...
Text Solution
|
If f(x)={{:(,(|x+2|)/(tan^(-1)(x+2)),x ne -2),(,2, x=-2):}, then f(x) ...
Text Solution
|
Let f(x)=(x+|x|)|x| . Then, for all x f is continuous
Text Solution
|
The set of all points where the function f(x)=sqrt(1-e^(-x^2)) is di...
Text Solution
|
The function f(x)=e^(-|x|) is continuous everywhere but not differe...
Text Solution
|
The function f(x)=[cos x] is
Text Solution
|
If f(x)=sqrt(1-sqrt(1-x^2)) , then f(x) is (a) continuous on [-1, 1] ...
Text Solution
|
If f(x) = sin ^(-1)((2x)/(1 + x^(2))) then f (x) is differentiable in ...
Text Solution
|
about to only mathematics
Text Solution
|
If f(x)=|x-a|varphi(x), where varphi(x) is continuous function, then f...
Text Solution
|
If f(x)=x^2+(x^2)/(1+x^2)+(x^2)/((1+x^2)^2)+. . . . +(x^2)/((1+x^2)^n)...
Text Solution
|
If f(x)= | log10x| then at x=1.
Text Solution
|
If f(x)=|log(e) x|,then
Text Solution
|
If f(x)=|log(e)|x||," then "f'(x) equals
Text Solution
|
Let f(x)={1/(|x|)\ \ \ \ \ for\ |x|geq1a x^2+b\ \ \ \ \ \ \ \ for\ |x|...
Text Solution
|
Let h(x)="min "{x,x^(2)} for every real number of x. Then, which one o...
Text Solution
|