If `f(x) = 0` has a root between `a and b` then `f(a) and f(b)` are of signs.

Assertion (A): For 0ltaltbltc equation (x-a)(x-b)-c=0 has no roots in (a,b) , Reason (R):For a continuous function f(x) equation f\'(x)=0 has at least one root between a and b if f(a) and f(b) are equal. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not the correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

Assertion (A): Equation (x-a)(x-b)-2=0, altb has one root less than a and other root greater than b. , Reason (R): A polynomial equation f\'(x)=0 has even number of roots between a and b if f(a) and f(b) have opposite signs. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not the correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

Assertion (A): Equation (x-p)(x-q)-r=0 where p,q,r epsilon R and 0ltpltqltr has roots in (p,q) , Reason(R): A polynomial equation f(x)=0 has odd number of roots between a and b (altb) if f(a) and f(b) have opposite signs (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not the correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

f(x)=int_a^x g(t)dt and f(x) has 5 roots between (a,b) . Find the number of roots of g(x)g'(x) between (a,b)

Let f (x) =ax ^(2) +bx + c,a ne 0, such the f (-1-x)=f (-1+ x) AA x in R. Also given that f (x) =0 has no real roots and 4a + b gt 0. Let p =b-4a, q=2a +b, then pq is:

If f(x) is twice differentiable and continuous function in x in [a,b] also f'(x) gt 0 and f ''(x) lt 0 and c in (a,b) then (f(c) - f(a))/(f(b) - f(a)) is greater than

If f(x)=x has non real roots, then the equation f(f(x))=x (A) has all real and unequal roots (B) has some real and non real roots (C) has all real and equal roots (D) has all non real roots

Statement-1: The equation (pi^(e))/(x-e)+(e^(pi))/(x-pi)+(pi^(pi)+e^(e))/(x-pi-e) = 0 has real roots. Statement-2: If f(x) is a polynomial and a, b are two real numbers such that f(a) f(b) lt 0 , then f(x) = 0 has an odd number of real roots between a and b.