f^(prime)(x)=varphi^(prime)(x)=f(x) for ...

`f^(prime)(x)=varphi^(prime)(x)=f(x)` for all `xdot` Also, `f(3)=5a n df^(prime)(3)=4.` Then the value of `[f(10)]^2` is _____

Text Solution

Verified by Experts

`(d)/(dx){[(f(x)]^(2)-[phi(x)]^(2)}`
`=2[f(x)cdotf'(x)-phi(x)cdotphi(x)]`
`=2[f(x)cdotphi(x)-phi(x)cdotf(x)][becausef'(x)=phi(x) and phi'(x)=f(x)]=0`
`"or "[f(x)]^(2)-[phi(x)]^(2)`= constant
`therefore" "[f(10)]^(2)-[phi(10)]^(2)=[f(3)]^(2)-[phi(3)]^(2)=[f(3)]^(2)-[f'(3)]^(2)`
`=25-16=9.`

Topper's Solved these Questions

DIFFERENTIATION
CENGAGE|Exercise JEE Previous Year|15 Videos
DIFFERENTIATION
CENGAGE|Exercise Multiple Correct Answers Type|3 Videos
DIFFERENTIATION
CENGAGE|Exercise Exercise (Matrix)|5 Videos
DIFFERENTIAL EQUATIONS
CENGAGE|Exercise Question Bank|7 Videos
DOT PRODUCT
CENGAGE|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

f^(prime)(x)=phi(x) and phi^(prime)(x)=f(x) for all x.Also,f(3)=5 and f^(prime)(3)=4 . Then the value of [f(10)]^2-[phi(10)]^2 is _____

If function f satisfies the relation f(x)xf^(prime)(-x)=f(-x)xf^(prime)(x)fora l lx ,a n df(0)=3,a n diff(3)=3, then the value of f(-3) is ______________

The function f: RvecR satisfies f(x^2)dotf^(x)=f^(prime)(x)dotf^(prime)(x^2) for all real xdot Given that f(1)=1 and f^(1)=8 , then the value of f^(prime)(1)+f^(1) is 2 b. 4 c. 6 d. 8

Let f: R->R be a differentiable function with f(0)=1 and satisfying the equation f(x+y)=f(x)f^(prime)(y)+f^(prime)(x)f(y) for all x ,\ y in R . Then, the value of (log)_e(f(4)) is _______

Suppose that f(x) is differentiable invertible function f^(prime)(x)!=0a n dh^(prime)(x)=f(x)dot Given that f(1)=f^(prime)(1)=1,h(1)=0 and g(x) is inverse of f(x) . Let G(x)=x^2g(x)-x h(g(x))AAx in Rdot Which of the following is/are correct? G^(prime)(1)=2 b. G^(prime)(1)=3 c. G^(1)=2 d. G^(1)=3

Suppose f(x)=e^(a x)+e^(b x) , where a!=b , and that f prime prime(x)-2f^(prime)(x)-15f(x)=0 for all xdot Then the value of (|a b|)/3 is ___

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot

If graph of y=f(x) is symmetrical about the point (5,0) and f^(prime)(7)=3, then the value of f^(prime)(3) is__________

Let f ''(x) be continuous at x=0 If ("lim")_(xvec0)(2f(x)-3a(f(2x)+bf(8x))/(sin^2x) exists and f(0)!=0,f^(prime)(0)!=0, then the value of (3a)/b is____

A nonzero polynomial with real coefficient has the property that f(x)=f^(prime)(x)dot f^(prime prime)(x)dot If a is the leading coefficient of f(x), then the value of 1/(2a) is____

CENGAGE-DIFFERENTIATION-Exercise (Numerical)

f^(prime)(x)=varphi^(prime)(x)=f(x) for all xdot Also, f(3)=5a n df^(p...
Text Solution
|
If y=f(x) is an odd differentiable function defined on (-oo,oo) such t...
Text Solution
|
If x^3+3x^2-9x=c is of the form (x-alpha)^2(x-beta) , then c is equal ...
Text Solution
|
If graph of y=f(x) is symmetrical about the point (5,0) and f^(prime)(...
Text Solution
|
Let g(x)=f(x)sinx ,w h e r ef(x) is a twice differentiable function on...
Text Solution
|
Let f(x)=(x-1)(x-2)(x-3)(x-n),n in N ,a n df(n)=5040. Then the value ...
Text Solution
|
y=f(x) , where f satisfies the relation f(x+y)=2f(x)+x y(y)+ysqrt(f(x)...
Text Solution
|
If function f satisfies the relation f(x)xf^(prime)(-x)=f(-x)xf^(prime...
Text Solution
|
If y=(a+b x^(3/2))/(x^(5/4))a n dy^(prime)=0a tx=5, then the value o...
Text Solution
|
Prove that (2^((log)2 1/4x)-3^(log)(27)(x^2+1)^3-2x >)/(7^(4(log)(49)...
Text Solution
|
(lim)(hvec0)((e+h)^(1n(e+h))-e)/hi s
Text Solution
|
If the function f(x)=-4e^((1-x)/2)+1+x+(x^2)/2+(x^3)/3a n dg(x)=f^(-1)...
Text Solution
|
Suppose that f(0)=0a n df^(prime)(0)=2, and let g(x)=f(-x+f(f(x)))dot ...
Text Solution
|
f(x) is polynomial of degree 4 with real coefficients such that f(x)=0...
Text Solution
|
A nonzero polynomial with real coefficient has the property that f(x)=...
Text Solution
|
A function is reprersented parametrically by the equations x=(1+t)/(t^...
Text Solution
|
Let z=(cosx)hat5a n dy="sin"xdot Then the value of 2(d^2z)/(dy^2)a tx=...
Text Solution
|
Let g(x)={(x^2+xtanx-xtan2x)/(a x+tanx-tan3x),x!=0 0,x=0 If g^(prime)...
Text Solution
|
Let f(x)=x+1/(2x+1/(2x+1/(2x+oo))) Compute the value of f(50)dotf^(p...
Text Solution
|
Let F(x)=f(x)g(x)h(x) for all real x ,w h e r ef(x),g(x),a n dh(x) are...
Text Solution
|