MHT CET202311 May 2023Evening ShiftMathematicsFunctionsActual

For all real x , the minimum value of 1-x+x^2 1+x+x^2 is

Options

A0
B1
C1 3
D3

Correct answer

C. 1 3

Step-by-step solution

aligned & f(x)= 1-x+x^2 1+x+x^2 & f^ (x)= (1+x+x^2 )(-1+2 x)- (1-x+x^2 )(1+2 x) (1+x+x^2 )^2 = & (-1+2 x-x+2 x^2-x^2+2 x^3 ) (1+x+x^2 )^2 = & -2+2 x^2 (1+x+x^2 )^2 aligned If f ^ (x)=0 , then -2+2 x^2 (1+x+x^2 )^2 =0 x^2=1 x= 1 f (x) at x=1 is 1 3 and f (x) at x=-1 is 1 , Minimum value of f (x) is 1 3 .

Practice Functions on Quantrex Academy →

More from Functions

If f : R A , defined by f ( x )= x + 3 x -1 , is an onto function then A = 2025 Let g(x)=1+x-[x] and f(x)= cases -1, & x 0 cases ([x] ) denotes the greatest integer less than or equal to (x ). Then for all x , f ( g ( x ))= 2025 The domain of the real valued function f(x)= 3 4-x^2 + ₁₀ (x^3-x ) is 2025 A real valued function f: A B defined by f(x)= 4-x^2 4+x^2 x A is a bijection. If -4 A then A B= 2025 Let [ x ] represent the greatest integer less than or equal to x , x = x -[ x ] , 2 =1.414 and 3 =1.732 . If f(x)= x+ [ x 1+x^2 ] is a real valued function, then f ( 2 )+ f (- 3 )= 2025 If the range of the function f(x)=-3 x-3 is 3,-6,-9,-18 , then which one of the following is not in the domain of f ? 2025 The domain of the function f(x)= _e ( 1 x^2-4 x+4 ) + ⁻¹ (x^2-2 ) is 2025 Consider the following statements Statement-I : A function f : A B is said to be one-one if and only if f(x) f(y) x y Statement-II : A relation f: A B is said to be a function if x 2025 Full Functions list All MHT CET PYQs