For any quadratic eqn px²+qx+r over [ a, a+h ] , then value of c belonging to (a, a+h) in Lagrange's Mean value theorm is

(a) dependent on p,q,r (b) independent of p but depends upon q,r

(c) Independent of p,q but depends on r (d) independent of p,q,

Lagrange Mean Value Theorem:

Let f : [a,b] in R be a continuous function such that f : (a, b) in R is differentiable. Then, there exists c  in (a, b) such that
                    f ' (c) = [ f(b) − f(a)] / [b − a]

so f '(c) = 2cp + q

f(a+h) = pa²+ph²+2pah+aq+qh+r

f(a)=pa²+aq+r

f(a+h) - f(a) = ph²+2pah+qh

so f ' (c)= ph+2pa+q = 2cp+q

so , c = a + (h/2)

c is independent of p,q,r

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Ramesh
IIT Kgp - 05 batch