Paragraph: Let f : [ 0 , 1 ] → R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f ( 0 ) = f ( 1 ) = 0 and satisfies f ′′ ( x ) − 2 f ′ ( x ) + f ( x ) ≥ e

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Paragraph: Let f : [ 0 , 1 ] → R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f ( 0 ) = f ( 1 ) = 0 and satisfies f ′′ ( x ) − 2 f ′ ( x ) + f ( x ) ≥ e x , x ∈ [ 0 , 1 ] . Question: If the function e − x f ( x ) assumes its minimum in the interval [ 0 , 1 ] at x = 4 1 ​ , which of the following is true?

TestHub · Accepted Answer

ϕ ′ x < 0 , x ∈ 0 , 1 / 4 and ϕ ′ x > 0 , x ∈ 1 / 4 , 1 ⇒ e - x f ′ x - e - x f x < 0 , x ∈ 0 , 1 / 4 f ′ x < f x , 0 < x < 1 / 4

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

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