Home
Class 12
MATHS
Let f(x) = a x^2 + bx + c, where a, b, c...

Let `f(x) = a x^2 + bx + c`, where `a, b, c in R, a!=0`. Suppose `|f(x)| leq1, x in [0,1]`, then

A

a+c

B

`|b|lel8`

C

`|c|le1`

D

`|a|+|b|+|c|le17`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
We have `f(x)=ax^(2)+bx+c`
`a,b,c epsionR[ :' a!=0]`
On puttig `x=0,1,1/2`, we get
`|c|le1`
`|a+b+c|le1`
and `|1/4a+1/2b+c|le1`
`implies-1lecle1`,
`-1lea+b+cle1`
and `-4lea+2b+4cle4`
`implies-4le4a+4b+4cle4`
and `-4le-a-2b-4cle4`
On adding we get
`-8le3a+2ble8`
Also `-8lea+2ble8`
`:.-16le2ale16`
`implies|a|le8`
`:'-1le-cle1,-8le-ale8`
We get `-16le2ble16`
`implies|b|le8`
`:.|a|+||+|c|le17`
Promotional Banner

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise Exercise (Passage Based Questions)|21 Videos

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise Exercise (Single Integer Answer Type Questions)|10 Videos

  • THEORY OF EQUATIONS

    ARIHANT MATHS|Exercise SCQ_TYPE|1 Videos

  • THE STRAIGHT LINES

    ARIHANT MATHS|Exercise The Straight Lines Exercise 8 : (Questions Asked in Previous 13 years Exams)|1 Videos

  • THREE DIMENSIONAL COORDINATE SYSTEM

    ARIHANT MATHS|Exercise Three Dimensional Coordinate System Exercise 12 : Question Asked in Previous Years Exam|2 Videos

Similar Questions

Explore conceptually related problems

Let f(x) = ax^2 + bx +C,a,b,c in R .It is given |f(x)|<=1,|x|<=1 The possible value of |a + c| ,if 8/3a^2+2b^2 is maximum, is given by

If f(x)=x^3+bx^2+cx+d and 0< b^2 < c , then

Let f(x) = Max. {x^2, (1 - x)^2, 2x(1 - x)} where x in [0, 1] If Rolle's theorem is applicable for f(x) on largest possible interval [a, b] then the value of 2(a+b+c) when c in [a, b] such that f'(c) = 0, is

Let f(x)=int_(-x)^(x)(t sin at + bt +c) dt, where a,b,c are non-zero real numbers , then lim_(x rarr0) (f(x))/(x) is

Verify Mean Value Theorem, if f(x) = x^3-5x^2-3x , in the interval [a, b], where a = 1 and b = 3. Find all c in (1,3) for which f'(c) = 0 .

Let f : R rarr R be a differentiable function at x = 0 satisfying f(0) = 0 and f'(0) = 1, then the value of lim_(x to 0) (1)/(x) . sum_(n=1)^(oo)(-1)^(n).f((x)/(n)) , is a. 0 b. -log2 c. 1 d. e

Let f(x)=ab sin x+bsqrt(1-a^(2))cosx+c , where |a|lt1,bgt0 then a. maximum value of f(x) is b, if c = 0 b. difference of maximum and minimum value of f(x) is 2b c. f(x)=c, if x=-cos^(-1)a d. f(x)=c, if x=cos^(-1)a

If f(x) = ax^(2) + bx + c is such that |f(0)| le 1, |f(1)| le 1 and |f(-1)| le 1 , prove that |f(x)| le 5//4, AA x in [-1, 1]

Suppose f(x)=e^(ax)+e^(bx)," where " a ne b, and that f''(x) -2f'(x)-15f(x)=0 for all x. Then the product ab is