Q. 785.0( 1 Vote )

# Using mean value theorem, prove that there is point on the curve y = 2x^{2} – 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 = 2x^{2} – 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 = 2x^{2} – 5x + 3

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

⇒ y = 2x^{2} – 5x + 3 is continuous at x ∈ [1,2]

Hence, condition 1 is satisfied.

Condition 2:

y = 2x^{2} – 5x + 3

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

y’ = 4x – 5 …(i)

⇒ y = 2x^{2} – 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 = 2x^{2} – 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 = 2x^{2} – 5x + 3, we have

⇒ y = 0

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

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