Let `A(x_1,y_1) and B(x_2,y_2)` be two points on the curve `f(x)= ax^2 + bx+c.` Using Lagrange's theorem prove that `EE` a point `C(x_2, y_3 )` on the curve where the tangent is parallel to AB. Also, show that `x_3=(x_1+x_2)/2.`

If A(x_1,y_1) an B(x_2,y_2) be two points on the curve y=ax^2+bx+c then Lagrange's mean value theorem show that there will be at least one point C(x_3,y_3) where the tangent will be parallel to the chord AB, Also show that x_3=(x_1+x_2)/2

The points on the curve x^2=3-2y , where the tangent is parallel to x+y=2, is

Using Rolle's theorem find the point on the curve y=x^(2)+1,-2lexle2 where the tangent is parallel to x-axis.

Find the points on the curve 2a^2y=x^3-3a x^2 where the tangent is parallel to x-axis.

Find the points on the curve 2a^2y=x^3-3a x^2 where the tangent is parallel to x-axis.

Find the point on the curve y^2-x^2 + 2x-1 = 0 where the tangent is parallel to the x-axis.

A(x_(1),y_(1)) and B(x_(2),y_(2)) are any two distinct points on the parabola y=ax^(2) +bx+c. if P(x_(3),y_(3)) be the point on the are AB where the tangent is parallel to the chord AB, then