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|25 Videos
DOT PRODUCT
CENGAGE|Exercise DPP 2.1|15 Videos

Similar Questions

Explore conceptually related problems

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

Given f'(x)=g(x)" and "g'(x)=f(x)." Also, "f(3)=5" and "f'(3)=4." Then, the value of: "[f(10)]^(2)-[g(10)]^(2)=

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

Let the function f satisfies f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) foe all x and f(0)=3 The value of f(x).f^(prime)(-x)=f(-x).f^(prime)(x) for all x, is

Let f(x+y)=f(x)f(y) for all x and y . Suppose f(5)=2 and f^(prime)(0)=3 , find f^(prime)(5) .

Q.5 If f(g(x))=g(f(x))=x for all real numbers x, and f(2)=5 and f(5)=3, then the value g(3)+g(f(2)) is

If f(g(x)) = g(f(x)) = x for all real numbers x, and f(2) = 5 and f(5) = 3, then the value of g(3)+ g(f(2)) is

Suppose that f is differentiable for all x and that f^(prime)(x)lt=2fora l lxdot If f(1)=2a n df(4)=8,t h e nf(2) has the value equal to 3 (b) 4 (c) 6 (d) 8

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 ...
Text Solution
|
If x^3+3x^2-9x+c is of the form (x-alpha)^2(x-beta) then c is equal to
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^(log2^(1//4)x)-3^(log(27)(x^(2)+1)^(3))-2x)/(7^(4log(49)...
Text Solution
|
lim(h to 0) ((e+h)^(In(e+h))-e)/(h) is-
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)=0 and f'(0)=2, and let g(x)=f(-x+f(f(x))). The value...
Text Solution
|
f(x) is polynomial of degree 4 with real coefficients such that f(x)=0...
Text Solution
|
A nonzero polynomial with real coefficients 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))). Then the v...
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
|