Let `f(x)=a x^2+b x+cdot` Consider the following diagram. Then Fig `c<0` `b >0` `a+b-c >0` `a b c<0`

Let f(X) = ax^(2) + bx + c . Consider the following diagram .

Let f(x) = ax^(2) - bx + c^(2), b ne 0 and f(x) ne 0 for all x in R . Then

If fig shows the graph of f(x)=a x^2+b x+c ,t h e n Fig a. c 0 c. a b >0 d. a b c<0

Let the parabolas y=x(c-x)a n dy=x^2+a x+b touch each other at the point (1,0). Then (a) a+b+c=0 (b) a+b=2 (c) b-c=1 (d) a+c=-2

Let f(x)=ax^2-bx+c^2, b != 0 and f(x) != 0 for all x ∈ R . Then (a) a+c^2 2b (c) a-3b+c^2 < 0 (d) none of these

Solve the following determinant equation: |[x+a, b, c], [c, x+b, a], [a ,b, x+c]|=0

Let f(x)=(1+b^2)x^2+2b x+1 and let m(b) the minimum value of f(x) as b varies, the range of m(b) is (A) [0,1] (B) (0,1/2] (C) [1/2,1] (D) (0,1]

Let f(x) be a continuous function such that f(a-x)+f(x)=0 for all x in [0,a] . Then int_0^a dx/(1+e^(f(x)))= (A) a (B) a/2 (C) 1/2f(a) (D) none of these

Let f(x)=a+b|x|+c|x|^(2) , where a,b,c are real constants. The, f'(0) exists if

Let f(x) be a function such that its derovative f'(x) is continuous in [a, b] and differentiable in (a, b). Consider a function phi(x)=f(b)-f(x)-(b-x)f'(x)-(b-x)^(2) A. If Rolle's theorem is applicable to phi(x) on, [a,b], answer following questions. If there exists some unmber c(a lt c lt b) such that phi'(c)=0 and f(b)=f(a)+(b-a)f'(a)+lambda(b-a)^(2)f''(c) , then lambda is