Using mean value theorem, prove that there is a 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 `AB`. Also, find that point.

Answer

N/a

Explanation

We have, `y = 2x^(2) - 5x + 3` . Which is continuous in `[1,2]` as it is a polynomial function.
Also, ` y' - 4x - 5`. Which exists in `(1,2)`.
By mean theorem , `3 c in (1,2)` at which drawn tangent is parallel to the chord `AB`,
`:. f'(c) = (f(2)- f(1))/(2-1)`
`rArr 4c - 5 = ((8-10+3)-(2-5+3))/(1)`
`rArr 4c - 5 = 1`
`:. c = 6/4 = 3/2 in (1,2)`
For `x = 3/2`, `y = 2(3/2)&(2) - 5 (3/2) + 3`
` = 2 xx 9/4 - 15/2 + 3 = (9-15+6)/(2) = 0`
Hence, `(3/2,0)` is the point on the curve `y = 2x^(2) - 5x+3` between the points `A (1,0) ,B(2,1)` , where tangent is parallel to the chord AB.