Using Lagrange's Mean Value theorem , find the co-ordinates of a point on the curve `y = x^(2)` at which the tangent drawn is parallel to the line joining the points (1,1) and (3,9).

Explanation

Let f(x) = `x^(2)` .
Here given interval is [1,3].
(i) The function f(x) has definite and unique value in [1,3]
`therefore` f(x) is continuous in [1,3] .
(ii) f'(x) = 2x
Which is defined in ]1,3[.
`therefore` f(x) is differentiable in ]1,3[ .
Thus the conditions of lagrange's Mean Value theorem satisfies .
Now there exists at least one value of `c in ]1,3[` such that
`f'(c) = (f(3) - f(1))/(3-1)`
`implies " " 2c = (9-1)/(2) = 4`
`implies " " c = 2 in ]1 , 3[`
Hence Lagrange's Mean Value theorem verified .
`therefore " " f(c) = c^(2)`
`implies " " f(2) = 2^(2) = 4`
`therefore " " ` The required point = (2,4). `" "` Ans.