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Question
Find a point on the parabola y = (x − 4)2, where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?
Solution
Let:
\[f\left( x \right) = \left( x - 4 \right)^2 = x^2 - 8x + 16\]
The tangent to the curve is parallel to the chord joining the points \[\left( 4, 0 \right)\] and \[\left( 5, 1 \right)\] .
Assume that the chord joins the points
So, \[x^2 - 8x + 16\] is continuous on \[\left[ 4, 5 \right]\] and differentiable on \[\left( 4, 5 \right)\] .
Consequently, there exists \[c \in \left( 4, 5 \right)\] such that
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