Problem 75 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)+f(y)+f(z)+f(x) \cdot f(y) \cdot f(z)=14\) for all \(x, y, z \in R\), then (A) \(f(0)=2\) (B) \(f^{\prime}(x)=0\), for all \(x \in R\) (C) \(f^{\prime}(x)>0\), for all \(x \in R\) (D) None of these

Step-by-Step Solution

Answer

Both (A) and (B) are correct. \(f(0) = 2\) and \(f'(x) = 0\).

Step 1: Understand the Function

Given the equation, \(f(x) + f(y) + f(z) + f(x) \cdot f(y) \cdot f(z) = 14\), identify that this equation must hold for all real numbers \(x, y, z\). We aim to determine an expression or characteristics for \(f(x)\).

Step 2: Assigning Specific Values

Test the equation by choosing specific values for \(x, y,\) and \(z\). Let's set \(x = y = z = 0\). Substitute these values into the equation: \(3f(0) + f(0)^3 = 14\).

Step 3: Simplify for \(f(0)\)

The equation \(3f(0) + f(0)^3 = 14\) can be solved by letting \(f(0) = a\). The equation becomes \(3a + a^3 = 14\). We need to find \(a\).

Step 4: Solve \(3a + a^3 = 14\)

Solve the cubic equation \(a^3 + 3a - 14 = 0\). By inspecting reasonable values, substituting \(a = 2\) results in \(8 + 6 - 14 = 0\). Therefore, \(f(0) = 2\).

Step 5: Determine Functional Form

To see if \(f(x)\) is constant, consider its derivative \(f'(x)\). Testing other values such as \(x = 1, y = 0, z = 0\) into the original equation should yield the same value; therefore, \(f(x) = 2\) for all \(x\). Thus, \(f'(x) = 0\).

Step 6: Check the Options

From \(f(x) = 2\), we find that no matter what \(x\) is, the function value is constant, satisfying both (A) \(f(0) = 2\) and (B) \(f'(x) = 0\). Clearly, the function is not increasing strictly, so (C) and (D) are incorrect.

Key Concepts

Constant Function

A constant function is a simple type of function where the output value is the same no matter what the input value is. Mathematically, if we have a function \( f(x) \), it is constant if \( f(x) = c \) for all \( x \), where \( c \) is a constant value. In the case of this exercise, we discovered that \( f(x) = 2 \) falls under this category.

- **Definition**: The main characteristic of a constant function is that the slope is zero. This results in a derivative of zero, \( f'(x) = 0 \).
- **Graph Characteristics**: The graph of a constant function is a horizontal line, indicating that no matter the value of \( x \), the \( y \) value remains the same.

Constant functions have no intervals of increase or decrease, they do not have any turning points, and are therefore easy to analyze. This simplicity makes them a crucial part of understanding basic function properties.

Cubic Equation

A cubic equation is a polynomial equation of degree three, typically in the form of \( ax^3 + bx^2 + cx + d = 0 \). Solving cubic equations is a bit more complex than linear or quadratic equations due to the higher degree, meaning there might be up to three real solutions.

- **General Form**: In the structure \( ax^3 + bx^2 + cx + d = 0 \), \( a, b, \) and \( c \) are coefficients, and \( d \) is the constant term.
- **Solution Methods**: Various methods exist for solving cubic equations, including factorization, synthetic division, and the use of the cubic formula. Inspection or guesswork for integer solutions can also prove effective, as seen when \( a = 2 \) was used, and the solution \( f(0) = 2 \) satisfied the equation \( 3a + a^3 = 14 \).

By substituting the guessed value into the equation, checking if the left side equates to the right side verifies the solution. Trial and error, especially with integers that make the components of the equation more manageable, can often lead to a solution quickly and easily.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It's essentially the slope of the function at any given point and indicates the rate of change or the sensitivity of one variable with respect to another.

- **Definition**: For a function \( f(x) \), the derivative \( f'(x) \) gives the slope of the tangent to the function at each point \( x \).
- **Connection to Constant Functions**: For constant functions like \( f(x) = 2 \), the derivative is always zero: \( f'(x) = 0 \). This is because the function does not change, resulting in a horizontal line which inherently has a slope of zero.

Understanding derivatives is crucial for analyzing and predicting behavior in various contexts, despite the simplicity of derivative calculations for constant functions.