Let ƒ(x) and g(x) be defined and differentiable for x ≥ x 0 and ƒ(x 0 ) = g(x 0 ),ƒ ′ (x) > g ′ (x) for x > x 0 , then -
Text Solution
Verified by ExpertsC
Consider the function φ (x) = ƒ(x) – g(x). On the interval [x 0 , x]. Then, φ (x) satisfies all the conditions of Lagrange’s mean value theorem on [x 0 , x].
∴ There exists at least one c ∈ (x 0 , x) such that
φ (x) – φ (x 0 ) = φ ′ (x – x 0 )
⇒ φ (x) = φ ′ (x – x 0 ) ( φ (x 0 ) = 0) … (1)
Also, φ′ (x) = ƒ ′ (x) – g ′ (x) ⇒ φ ′ = ƒ ′ – g ′ > 0
( ƒ ′ (x) > g ′ (x) for x > x 0 )
∴ from (1), φ (x) > 0, for x > x 0
⇒ ƒ(x) – g(x) > 0 for x > x 0
or ƒ(x) > g(x) for x > x 0 .
Prepare Smarter with CGP Edu
Get practice questions, solutions, and test series in one place.
Write a Review
Share your experience with this question and solution.
Commentary
Send your comment, doubt, correction, or feedback to admin.
Similar Questions
Explore conceptually related problems