Problem 7 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

Let \(f(x)=x^{3}+b x^{2}+c x+d, 0

Step-by-Step Solution

Answer

The function is increasing (option b).

Step 1: Understanding the Function

We start with the given function \( f(x) = x^3 + bx^2 + cx + d \). This is a cubic polynomial, which is a continuous function across all real numbers.

Step 2: Analyzing Stationary Points

To determine where the function might be stationary, we need to find where its derivative \( f'(x) = 3x^2 + 2bx + c \) equals zero. Solving this quadratic equation for \( x \) gives possible stationary points.

Step 3: Studying the Function's Behavior

Analyze the inequality \( 0 < b^2 < c \). This condition implies that the quadratic in the derivative \( 3x^2 + 2bx + c \) has a positive discriminant \( 4b^2 - 12c < 0 \), which can mean the derivative has no real roots, and the derivative never passes through zero, suggesting the derivative never changes sign.

Step 4: Concluding with Monotonicity

Since \( f'(x) \) is positive for all \( x \) due to no real roots in the derivative equation, the function \( f(x) \) is strictly increasing, as its slope does not change sign.

Key Concepts

Derivative of a Function

The derivative of a function is a fundamental concept in calculus. It measures how the function changes at any given point. For a cubic polynomial like \( f(x) = x^3 + bx^2 + cx + d \), the derivative \( f'(x) \) is obtained by applying power rules to each term:

• The derivative of \( x^3 \) is \( 3x^2 \).
• The derivative of \( bx^2 \) is \( 2bx \).
• The derivative of \( cx \) is \( c \).
• The derivative of a constant \( d \) is zero.

Combining these, we get the derivative of the function as \( f'(x) = 3x^2 + 2bx + c \). This quadratic expression in \( x \) is critical for understanding where the function might be increasing or decreasing, as it represents the slope of \( f(x) \) at any point \( x \).

Stationary Points

Stationary points in the context of calculus are points where the derivative of a function is zero. For our cubic polynomial, we find stationary points by setting the derivative \( f'(x) = 3x^2 + 2bx + c \) equal to zero:\[ 3x^2 + 2bx + c = 0 \]Solving this quadratic equation gives us potential values of \( x \) where the function may be stationary. However, an important condition given in the problem is \( 0 < b^2 < c \). This affects the discriminant of the quadratic equation \( 4b^2 - 12c \).Since \( 4b^2 - 12c \) is less than zero, this quadratic equation has no real roots. This means there are no real x-values making the derivative equal to zero, suggesting there are no stationary points on the real-number line for this particular function.

Monotonicity Analysis

Monotonicity refers to the property of a function to be consistently increasing or decreasing. For \( f(x) = x^3 + bx^2 + cx + d \), assessing monotonicity involves examining the sign of the derivative \( f'(x) = 3x^2 + 2bx + c \).Given the condition \( 0 < b^2 < c \), we find that the discriminant of the derivative's equation is negative \((4b^2 - 12c < 0)\). Thus, the quadratic expression \( 3x^2 + 2bx + c \) doesn’t have any real roots, implying it never crosses zero.Since the expression never evaluates to zero and does not change sign, it remains positive for all values of \( x \). As the derivative is always positive, the slope of the cubic polynomial is always upward, indicating that the function is strictly increasing across its entire domain, confirming its monotonicity as an increasing function.