One is "some others"

Calculus Level 3

Let f ( x ) f(x) be a quadratic equation which is positive for all real x x . If g ( x ) = f ( x ) + f ′ ( x ) + f ′ ′ ( x ) g(x) = f(x) + f'(x) + f''(x) for all real x x , what is g ( x ) g(x) ?

Clarification: f ′ ( x ) f'(x) is the derivative of f ( x ) f(x) and f ′ ′ ( x ) f''(x) is the derivative of f ′ ( x ) f'(x) .

Non-negative Non-positive Postive Negative

Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Tom Engelsman
Dec 29, 2018

Let f ( x ) = A x 2 + B f(x) = Ax^2 + B with A , B ∈ R + . A,B \in \mathbb{R^{+}}. The first and second derivatives compute to: f ′ ( x ) = 2 A x , f ′ ′ ( x ) = 2 A . f'(x) = 2Ax, f''(x) = 2A. The sum g ( x ) = f ( x ) + f ′ ( x ) + f ′ ′ ( x ) g(x) = f(x) + f'(x) + f''(x) evaluates to:

g ( x ) = ( A x 2 + B ) + ( 2 A x + 2 A ) = A ( x 2 + 2 x + 1 ) + ( A + B ) = A ( x + 1 ) 2 + ( A + B ) g(x) = (Ax^2 + B) + (2Ax + 2A) = A(x^2 + 2x + 1) + (A + B) = A(x+1)^2 + (A+B)

The minimum value of g ( x ) g(x) over all reals is A + B > 0 A+B > 0 ; hence, it is a positive-valued function.

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