Answer:
x = 1/2
Step-by-step explanation:
Solution:-
- Any line tangent to a curve defined by y = f(x) have the gradient -slope equal to the first derivative of the curve, dy/dx :
y = Ln ( 1 - x )
dy/dx = -1 / ( 1 - x )
- So the line tangent to the curve has slope defined by the first derivative evaluated, while the equation of tangent line is:
y = mx + c
m = dy/dx = -1 / ( 1 - x )
Where, c = y - intercept = 2
y = -x / ( 1 - x ) + 2 ..... Equation of tangent
- The x-point of intersection can be evaluated by simultaneously solving the equation of tangent and curve y = f(x):
-x + 2 - 2x = ( 1 - x )*Ln ( 1 -x )
-3x + 2 = Ln ( 1 -x ) - Ln ( 1 - x )^x
-3x + 2 = Ln ( 1 - x ) ^ ( 1 - x )
( 1 - x ) ^ ( 1 - x ) = e^ ( -3x + 2 )
1 - x = e , x = 1-e
1 - x = -3x + 2, x = 1/2
- The x-value of point of intersection is x = 1/2