Analyzing the Sign of g(x) for a Positive Quadratic f(x)

The problem asks us to determine the sign of the function \( g(x) = f(x) + f'(x) + f''(x) \) for any real number \( x \), given that \( f(x) \) is a quadratic expression which is positive for all real \( x \).

Properties of a Quadratic Expression Positive for all Real x

Let the quadratic expression \( f(x) \) be represented as \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are real coefficients.

For a quadratic expression \( f(x) \) to be positive for all real values of \( x \) (i.e., \( f(x) > 0 \) for all \( x \in \mathbb{R} \)), two necessary and sufficient conditions must be met:

• The leading coefficient \( a \) must be positive. So, \( a > 0 \).
• The discriminant of the quadratic equation \( ax^2 + bx + c = 0 \) must be negative. The discriminant is \( \Delta_f = b^2 - 4ac \). So, \( b^2 - 4ac < 0 \).

Calculating f'(x) and f''(x)

Next, we need to find the first and second derivatives of \( f(x) \).

The first derivative \( f'(x) \) is:

The second derivative \( f''(x) \) is:

Defining and Simplifying g(x)

Now, we can write the expression for \( g(x) \) by substituting \( f(x) \), \( f'(x) \), and \( f''(x) \):

Combining like terms, we get:

This expression for \( g(x) \) is also a quadratic in \( x \). Let's write it in the standard quadratic form \( Ax^2 + Bx + C \), where:

• \( A = a \)
• \( B = b + 2a \)
• \( C = c + b + 2a \)

So, \( g(x) = Ax^2 + Bx + C \).

Analyzing the Sign of g(x)

To determine the sign of the quadratic expression \( g(x) \) for all real \( x \), we need to examine its leading coefficient and its discriminant.

Leading Coefficient of g(x)

The leading coefficient of \( g(x) \) is \( A = a \). From the given information that \( f(x) > 0 \) for all real \( x \), we know that \( a > 0 \). Therefore, the leading coefficient of \( g(x) \) is positive:

Discriminant of g(x)

The discriminant of \( g(x) \) is \( \Delta_g = B^2 - 4AC \). Substituting the values of \( A, B, \) and \( C \):

Expanding the terms:

We know from the condition on \( f(x) \) that \( b^2 - 4ac < 0 \). This means \( b^2 - 4ac \) is a negative value.

Also, since \( a > 0 \), \( a^2 \) is positive, and thus \( 4a^2 \) is positive.

The discriminant of \( g(x) \) is \( \Delta_g = (b^2 - 4ac) - 4a^2 \). This is a negative value minus a positive value. A negative value minus a positive value is always negative.

Conclusion about g(x)

The quadratic expression \( g(x) = Ax^2 + Bx + C \) has:

• A positive leading coefficient (\( A > 0 \)).
• A negative discriminant (\( \Delta_g < 0 \)).

These are precisely the conditions for a quadratic expression to be positive for all real values of \( x \).

Therefore, \( g(x) > 0 \) for any real \( x \).

Summary of Findings

Since \( a > 0 \) and \( b^2 - 4ac < 0 \), we showed that the leading coefficient of \( g(x) \) is \( A = a > 0 \) and the discriminant of \( g(x) \) is \( \Delta_g = (b^2 - 4ac) - 4a^2 \), which is negative because it's a negative term minus a positive term.

Thus, \( g(x) \) is always positive for any real \( x \).

Revision Table - Quadratic Properties

Additional Information - Derivatives of Polynomials

The derivatives of a polynomial function \( P(x) = c_n x^n + c_{n-1} x^{n-1} + \dots + c_1 x + c_0 \) are found using the power rule of differentiation:

• The derivative of a term \( cx^k \) is \( kcx^{k-1} \).
• The derivative of a constant \( c \) is 0.

For a quadratic \( f(x) = ax^2 + bx + c \):

• \( f'(x) \): Apply the power rule to each term.
  • Derivative of \( ax^2 \) is \( 2ax^{2-1} = 2ax \).
  • Derivative of \( bx \) (which is \( bx^1 \)) is \( 1 \cdot bx^{1-1} = b x^0 = b \).
  • Derivative of \( c \) (a constant) is \( 0 \).
  So, \( f'(x) = 2ax + b \).
• \( f''(x) \): Differentiate \( f'(x) \).
  • Derivative of \( 2ax \) is \( 1 \cdot 2ax^{1-1} = 2a x^0 = 2a \).
  • Derivative of \( b \) (a constant) is \( 0 \).
  So, \( f''(x) = 2a \).

Understanding derivatives is crucial for analyzing the rate of change and properties of functions like quadratics.