Problem 131 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Given that \(f^{\prime}(x)>g^{\prime}(x)\) for all real \(x\) and \(f(0)=g(0)\), then (A) \(f(x)>g(x) \forall x \in(0, \infty)\) (B) \(f(x)g(x) \forall x \in(-\infty, 0)\)

Step-by-Step Solution

Answer

Option (A) is correct: \(f(x) > g(x)\) for \(x \in (0, \infty)\).

Step 1: Understand Given Conditions

We are given that the derivative of function \(f\), \(f'(x)\), is greater than the derivative of function \(g\), \(g'(x)\), for all real \(x\). This implies that as \(x\) increases, \(f(x)\) increases faster than \(g(x)\). Additionally, \(f(0) = g(0)\) is given, meaning both functions have the same value at \(x = 0\).

Step 2: Evaluate Function Behavior on Positive Real Line

For \(x > 0\), because \(f'(x) > g'(x)\), the rate of increase of \(f(x)\) is greater than that of \(g(x)\). Starting from \(f(0) = g(0)\), as \(x\) becomes greater than zero, \(f(x)\) will increase at a greater rate than \(g(x)\). This means \(f(x) > g(x)\) for all \(x \in (0, \infty)\).

Step 3: Evaluate Function Behavior on Negative Real Line

For \(x < 0\), the condition \(f'(x) > g'(x)\) means that \(f(x)\) is decreasing at a slower rate than \(g(x)\), or increasing at a faster rate if both are increasing. Since \(f(0) = g(0)\), it follows that as \(x\) goes negative, \(f(x) > g(x)\), resulting in \(f(x) < g(x)\). This implies \(f(x) > g(x)\) because \(f'(x) > g'(x)\), and thus \(f(x) > g(x)\) for \(x < 0\).

Step 4: Determine the Correct Statement

Based on the evaluations, the correct behavior of the functions is: \(f(x) > g(x)\) for all \(x \in (0, \infty)\) as concluded from behavior evaluations for positive \(x\) values and starting from the shared point at zero.

Key Concepts

Derivative

The derivative of a function provides us with valuable insights about the rate at which the function changes with respect to its variable. In this exercise, understanding the derivatives of functions \(f(x)\) and \(g(x)\) is crucial, as it tells us how rapidly each function is increasing or decreasing.

• When we say \(f'(x) > g'(x)\), it means that the slope of the tangent at any point \(x\) on \(f(x)\) is steeper than that on \(g(x)\). This informs us that with every small increase in \(x\), \(f(x)\) rises more quickly than \(g(x)\).
• Therefore, understanding derivatives allows us to predict the function behavior over intervals, informing us which function grows faster or slower compared to another.

By comparing the derivatives, we can conclusively determine changes in function behavior without having to manually calculate each value.

Real Numbers

Real numbers are a fundamental part of calculus and mathematical analysis. They include all the rational numbers like integers and fractions, as well as irrational numbers such as \( \pi \) and the square root of non-perfect squares. This comprehensive set is essential when analyzing and comparing functions.

• In this exercise, the concept of real numbers is important because the comparisons between \(f(x)\) and \(g(x)\) are made over the real line.
• We consider values from \(-\infty\) to \(\infty\) which implies an infinite, continuous spectrum of possibilities for \(x\).

The behavior of functions is analyzed within this entire domain, allowing for a complete understanding of their comparisons.

Function Behavior

Understanding function behavior involves analyzing how a function changes over its domain. In this particular instance, we look at how \(f(x)\) and \(g(x)\) behave relative to each other under certain conditions.

• For \(x > 0\), knowing that \(f'(x) > g'(x)\) implies that \(f(x)\) outpaces \(g(x)\) in terms of growth rate. Consequently, starting from the point where both functions equal at zero, \(f(x)\) will always be greater than \(g(x)\).
• Conversely, for \(x < 0\), this condition indicates that \(f(x)\) decreases slower than \(g(x)\), thus maintaining \(f(x) > g(x)\) throughout the negative \(x\) values.

By understanding these behavioral trends, we can make informed predictions about the comparison of these functions.

Mathematical Analysis

Mathematical analysis is the formal study of limits, differentiation, integration, infinite series, and other concepts dealing with continuous functions. In this critical branch of mathematics, analyzing and comparing functions based on their derivatives provides deep insights into their behavior.

• We utilize mathematical analysis to interpret derivatives and function values, understanding how two functions interact over an interval.
• In this exercise, the analysis reveals that due to \(f'(x) > g'(x)\) and \(f(0) = g(0)\), we can predict the relationship between \(f(x)\) and \(g(x)\) across the real numbers.

By applying these methods, we ensure precise and logical conclusions in understanding the dynamics of function behaviors.