Q. 785.0( 1 Vote )

Using mean value theorem, prove that there is point on the curve y = 2x2 – 5x + 3 between the points A(1, 0) and B(2, 1), where tangent is parallel to the chord A. Also, find that point.

Answer :

Given: y = 2x2 – 5x + 3 in [1,2]

Now, we have to show that f(x) verify the Mean Value Theorem


First of all, Conditions of Mean Value theorem are:


a) f(x) is continuous at (a,b)


b) f(x) is derivable at (a,b)


If both the conditions are satisfied, then there exist some ‘c’ in (a,b) such that



Condition 1:


y = 2x2 – 5x + 3


Since, f(x) is a polynomial and we know that, every polynomial function is continuous for all x R


y = 2x2 – 5x + 3 is continuous at x [1,2]


Hence, condition 1 is satisfied.


Condition 2:


y = 2x2 – 5x + 3


Since, f(x) is a polynomial and every polynomial function is differentiable for all x R


y’ = 4x – 5 …(i)


y = 2x2 – 5x + 3 is differentiable at [1,2]


Hence, condition 2 is satisfied.


Thus, Mean Value Theorem is applicable to the given function.


Now,


f(x) = y = 2x2 – 5x + 3 x [1,2]


f(a) = f(1) = 2(1)2 – 5(1) + 3 = 2 – 5 + 3 = 0


f(b) = f(2) = 2(2)2 – 5(2) + 3 = 8 – 10 + 3 = 1


Then, there exist c (0,1) such that



Put x = c in equation, we get


y’ = 4c – 5 …(i)


By Mean Value Theorem,





4c – 5 = 1


4c = 6



So, value of



Thus, Mean Value Theorem is verified.


Put in given equation y = 2x2 – 5x + 3, we have





y = 0


Hence, the tangent to the curve is parallel to the chord AB at


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