verify lagrange's mean value theorem for the follwing function f(x)=px2 + qx + r where p not equal to 0 and x belongs to [a,b]

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Since f(x)=px2+qx+r is everywhere continuous and differentiable being a polynomial,therefore f(x) is continuous on a,b and differentiable on a,b.Thus, both the conditionsof Lagrange's meaan value theorem are satisfied.So, there must exist at least one ca,b such thatf'(c)=fb-fab-aNow, f(x)=px2+qx+rf'(x)=2px+q and f(b)=pb2+qb+r and f(a)=pa2+qa+rf'(x)= fb-fab-a2px+q=pb2+qb+r-pa2-qa-rb-a2px+q=pb2-a2+qb-ab-a2px+q=b-apb+a+qb-a2px+q=pb+a+q2px=p(b+a)x=b+a2Thus, c=b+a2a,b such that f'(c)=fb-fab-aHence, Lagrange's mean value theorem is verified.
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