If varphi(x) is a polynomial function an...

If `varphi(x)` is a polynomial function and `varphi^(prime)(x)>varphi(x)AAxgeq1a n dvarphi(1)=0,` then `varphi(x)geq0AAxgeq1` `varphi(x)

`phi(x)ge0forallxge1`

`phi(x)lt0forallxge1`

`phi(x)=0forallxge1`

none of these

Text Solution

Verified by Experts

The correct Answer is:

Given `phi (x) -phi(x) gt0 forall x ge 1`
or `e^(-x){phi(x) -phi(x)}gt0 forall x ge 1`
or `(d)/(dx)e^(-x)phi(x) gt 0 forall x ge1`
Therefore `e^(-x) phi (x)` is an increasing function `forall x ge 1`
since `phi(x)` is a polynomial
`e^(-x)phi(x)gte^(-1)phi(1)or e^(-x)phi(x)gt0`
`or phi(x) gt0`

Topper's Solved these Questions

MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS
CENGAGE|Exercise Exercise (Multiple)|40 Videos
MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS
CENGAGE|Exercise Exercise (Comprehension)|42 Videos
MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS
CENGAGE|Exercise Exercise 6.7|5 Videos
METHODS OF DIFFERETIATION
CENGAGE|Exercise Question Bank|29 Videos
MONOTONOCITY AND NAXINA-MINIMA OF FUNCTIONS
CENGAGE|Exercise Comprehension Type|6 Videos

Similar Questions

Explore conceptually related problems

If varphi(x) is differentiable function AA x in R and a in R^(+) such that varphi(0)=varphi(2a),varphi(a)=varphi(3a)and varphi(0)!=varphi(a) then show that there is at least one root of equation varphi'(x+a)=varphi'(x)in(0,2a)

If y+x(dy)/(dx)=x(varphi(xy))/(varphi'(xy)) then varphi(xy) is equal to

If f(x)=|x-a|varphi(x), where varphi(x) is continuous function, then (a) f^(prime)(a^+)=varphi(a) (b) f^(prime)(a^-)=-varphi(a) (c) f^(prime)(a^+)=f^(prime)(a^-) (d) none of these

Evaluate: int_(pi/4)^((3pi)/4) varphi/(1-sinvarphi)d varphi

The solution of the differential equation (dy)/(dx)=(y)/(x)+(varphi((y)/(x)))/(varphi'((y)/(x))) is varphi((y)/(x))=kxbx varphi((y)/(x))=kc*varphi((y)/(x))=kyd.y varphi((y)/(x))=k

Let int e^(x){f(x)-f'(x)}dx=varphi(x) .Then int e^(x)f(x)dx is varphi(x)=

Evaluate: int(2sin2varphi-cosvarphi)/(6-cos^2varphi-4\ sinvarphi)\ dvarphi

CENGAGE-MONOTONICITY AND MAXIMA MINIMA OF FUNCTIONS-Exercise (Single)

Suppose that f is a polynomial of degree 3 and that f^(x)!=0 at any of...
Text Solution
|
A function g(x) is defined as g(x)=1/4f(2x^2-1)+1/2f(1-x^2)a n df(x) i...
Text Solution
|
If varphi(x) is a polynomial function and varphi^(prime)(x)>varphi(x)A...
Text Solution
|
If f''(x) gt forall in R, f(3)=0 and g(x) =f(tan^(2)x-2tanx+4y)0ltxlt(...
Text Solution
|
If f(x)=x+sinx ,g(x)=e^(-x),u=sqrt(c+1)-sqrt(c) v=sqrt(c) -sqrt(c-1),(...
Text Solution
|
The number of solutions of the equation x^3 +2 x^2 +5 x + 2 cos x = 0 ...
Text Solution
|
Let f(x)=cospix+10x+3x^2+x^3,-2lt=xlt=3. The absolute minimum value of...
Text Solution
|
The global maximum value of f(x0=(log)(10)(4x^3-12 x^2+11 x-3),x in [2...
Text Solution
|
f:R rarr R,f(x) is differentiable such that f(x)=k(x^(5)+x).(kne0) The...
Text Solution
|
The value of a for which the function f(x)=asinx+(1/3)sin3x has an ext...
Text Solution
|
If f(x)=alog|x|+b x^2+x has its extremum values at x=-1a n dx=2, then ...
Text Solution
|
If a function f(x) has f^(prime)(a)=0a n df^(a)=0, then x=a is a maxim...
Text Solution
|
The function f(x)=sin^4x +cos ^4 x increases if
Text Solution
|
If f(x) =sin^(4)x+cos^(4)x increases, if
Text Solution
|
If f(x)=x^5-5x^4+5x^3-10 has local maximum and minimum at x=pa n dx=q ...
Text Solution
|
Let P(x)=a0+a1x^2+a2x^4++an x^(2n) be a polynomial in a real variable ...
Text Solution
|
Let f(x) ={|x|for 0< |x| , 2 for x=0
Text Solution
|
If f(x)=x^3+b x^2+c x+d and 0<b^2<c , then f(x) is a strictly increasi...
Text Solution
|
If f(x)={(sin(x^(2)-3x),xle0 .^x+5x^(2),xgt0
Text Solution
|
The greatest value of f(x)=cos(x e^([x])+7x^2-3x),x in [-1,oo], is (wh...
Text Solution
|