Question
Question: On the curve ƒ(x) = x<sup>3</sup>, the point at which tangent line is parallel to the chord through ...
On the curve ƒ(x) = x3, the point at which tangent line is parallel to the chord through the points A(–1, –1) and B(2, 8) is –
A
(–1, 1)
B
(1, –1)
C
(–1, –1)
D
(1, 1)
Answer
(1, 1)
Explanation
Solution
In the interval [–1, 2], whose end points are the abscissas of the points A and B, the function ƒ(x) = x3 satisfies conditions of Lagrange’s mean value theorem. Therefore, there exists a point P on the arc AB at which tangent is parallel to the chord AB.
Now, by Lagrange’s mean value theorem,
ƒ(2) – ƒ(–1) = ƒ ′ (3) (2 – (–1))
⇒ 8 + 1 = 3c2 (3) ⇒ c = ± 1
These values of c are the abscissas of the desired points but c = –1 does not lie on (–1, 2). ∴ c = 1. Corresponding value of ordinate is (1)3 i.e. 1
∴ P is (1, 1).