Problem 29 - A Textbook of Differential Calculus

Question

for the function \(y=f(x)=\left(x^{2}+b x+c\right) e^{x}\), which of the following holds? (a) If \(f(x)>0\) for all real \(x \Rightarrow f^{\prime}(x)>0\) (b) If \(f(x)>0\) for all real \(x \Rightarrow f^{\prime}(x)>0\) (c) If \(f^{\prime}(x)>0\) for all real \(x \Rightarrow f(x)>0\) (d) If \(f^{\prime}(x)>0\) for all real \(x \neq f(x)>0\)

Step-by-Step Solution

Answer

The correct option is (c).

Step 1: Compute f'(x)

\(f(x) = (x^2 + bx + c)e^x\)

Using the product rule:
\(f'(x) = (2x + b)e^x + (x^2 + bx + c)e^x = [x^2 + (b+2)x + (b+c)]e^x\)

Step 2: Analyze condition f'(x) > 0 for all real x

Since \(e^x > 0\) for all \(x\), \(f'(x) > 0\) requires:
\(x^2 + (b+2)x + (b+c) > 0\) for all \(x\).

This holds iff the discriminant is negative:
\((b+2)^2 - 4(b+c) < 0 \implies b^2 + 4b + 4 - 4b - 4c < 0 \implies b^2 - 4c < -4\)

Step 3: Check whether this implies f(x) > 0

\(f(x) > 0\) for all \(x\) requires \(x^2 + bx + c > 0\) for all \(x\) (since \(e^x > 0\)).
This holds iff discriminant \(b^2 - 4c < 0\).

From Step 2, \(f'(x) > 0\) gives \(b^2 - 4c < -4 < 0\).
So \(b^2 - 4c < 0\), which implies \(f(x) > 0\) for all \(x\). \(\checkmark\)

Step 4: Verify option (a)/(b) fails and conclude

Conversely, \(f(x) > 0\) gives \(b^2 - 4c < 0\), but this does NOT imply \(b^2 - 4c < -4\), so \(f'(x) > 0\) does not necessarily follow from \(f(x) > 0\).

Counterexample: \(b = 0, c = 1\). Then \(f(x) = (x^2+1)e^x > 0\) but \(f'(x) = (x^2+2x+1)e^x = (x+1)^2 e^x \geq 0\), with \(f'(-1) = 0\), so \(f'(x)\) is not strictly positive everywhere.

The correct answer is \(\boxed{(c)}\): If \(f'(x) > 0\) for all real \(x\), then \(f(x) > 0\).

Key Concepts

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. In our function, this is represented by \( e^x \). A key property of exponential functions is that they grow rapidly and are always positive over the set of real numbers. No matter what real number you substitute for \( x \), the exponential function \( e^x \) will yield a positive result. This characteristic plays a significant role in many types of functions, especially when analyzing their behavior as \( x \) increases or decreases.

Polynomial

A polynomial is a sum of terms, each consisting of a variable raised to a non-negative integer exponent, multiplied by a coefficient. In the function \( y = (x^2 + bx + c) e^x \), the part \( x^2 + bx + c \) is the polynomial. This particular polynomial is quadratic because the highest power of \( x \) is 2. Quadratic polynomials have a parabolic shape when graphed, and they can open upwards or downwards depending on the leading coefficient (the coefficient of \( x^2 \)). The coefficients \( b \) and \( c \) adjust the position and shape of the parabola. Understanding the polynomial component of a function is crucial for predicting and analyzing the function's behavior.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. For the function \( f(x) = (x^2 + bx + c)e^x \), we calculate the derivative to learn about the function’s rates of change. The derivative of a product like this one can be found using the product rule, which states that the derivative of a product of two functions \( u \) and \( v \) is \( u'v + uv' \). Applying this rule to our function gives:

• Derivative of \( x^2 + bx + c \) is \( 2x + b \)
• Derivative of \( e^x \) is \( e^x \)

Hence, the derivative of \( f(x) \) is \((2x + b)e^x + (x^2 + bx + c)e^x\). Calculating derivatives is essential for function analysis, as it helps determine points where the function increases or decreases.

Function Analysis

Function analysis involves a detailed examination of a function's behavior. By studying the behavior of the function and its derivative, we can gain insights into the overall shape and direction of the graph of \( f(x) = (x^2 + bx + c)e^x \). With the derivative, you can identify critical points where the function changes from increasing to decreasing, or vice versa. These are points where \( f'(x) = 0 \). Additionally, by reviewing the sign of the derivative, we can determine intervals where the function is strictly increasing or decreasing. Understanding these properties helps us predict the function's behavior without having to graph it comprehensively.