Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If a differentiable function \(f(x)\) has a relative minimum at \(x=0\), then the function \(y=f(x)+a x+b\) has a relative minimum at \(x=0\) for (A) all \(a>0\) (B) all \(b>0\) (C) all \(a\) and \(b\) (D) all \(b\) if \(a=0\)

Step-by-Step Solution

Answer

(D) all \(b\) if \(a=0\)

Step 1: Analyze the Given Function

The problem states that the function \( f(x) \) has a relative minimum at \( x=0 \). This implies that \( f'(0) = 0 \) and potentially \( f''(0) > 0 \) to confirm the minimum.

Step 2: Consider Modified Function

We want to find if \( y = f(x) + ax + b \) also has a relative minimum at \( x=0 \). Consider \( y \) as the new function and analyze its behavior at \( x=0 \).

Step 3: Differentiate the New Function

Calculate the first derivative: \( y' = f'(x) + a \). At \( x=0 \), \( y'(0) = f'(0) + a = a \). Since \( f'(0) = 0 \), it simplifies to \( y'(0) = a \).

Step 4: Condition for Relative Minimum

For a relative minimum at \( x=0 \), we need \( y'(0) = 0 \) and \( y''(0) > 0 \). Since \( y'(0) = a \), we require \( a = 0 \) to satisfy the first condition.

Step 5: Second Derivative Check

If \( a = 0 \), then the second derivative \( y'' = f''(x) \). At \( x=0 \), \( y''(0) = f''(0) \). We know \( f''(0) > 0 \) for a relative minimum.

Step 6: Conclusion for Parameter Conditions

The function \( y = f(x) + ax + b \) has a relative minimum at \( x=0 \) if \( a = 0 \) and because \( f''(0) > 0 \), \( b \) has no effect. Thus, the answer is option (D).

Key Concepts

Differentiable function

A differentiable function is a fundamental concept in calculus, referring to a function that has a derivative at each point in its domain. This means that not only does the function exist and is continuous at every point, but it also has a well-defined tangent line at those points. Differentiability is crucial as it allows us to understand how a function changes infinitesimally, making it possible to apply various calculus techniques.For a function to be differentiable at a point, the limit:\[ \lim_{{h \to 0}} \frac{f(x+h) - f(x)}{h} \]must exist and be finite. This expression is known as the derivative of the function at that point. Differentiable functions exhibit smoothness, meaning they have no sharp corners or cusps.Key points to recall about differentiable functions:

• Continuity: A differentiable function must be continuous at every point in its domain.
• Tangential Line: It has a linear approximation or tangent line at each point.
• Smoothness: No abrupt changes or breaks in the graph of the function.

Relative minimum

A relative minimum of a function occurs at a certain point if the function's value at that point is less than or equal to the values at nearby points. It indicates a local dip in the graph of the function, where the function stops decreasing and starts increasing.Here's how to identify a relative minimum using calculus:

• You need to check both the first and second derivatives of the function.
• At a relative minimum, the first derivative, \( f'(x) \), equals zero, signifying a critical point where the slope of the tangent is flat.
• The second derivative, \( f''(x) \), should be greater than zero (\( f''(x) > 0 \)), indicating the concavity is upwards at that point, thus confirming a local minimum.

Recognizing the signs:- A positive second derivative shows the function is curving upwards, supporting the existence of a minimum.- A negative second derivative (\( f''(x) < 0 \)) would suggest a maximum, not a minimum.

Second derivative test

The second derivative test is a powerful tool in calculus used to determine the nature of critical points found using the first derivative. This test helps to differentiate between relative minima, maxima, and points of inflection.The steps are simple:

• First, find the critical points by setting the first derivative \( f'(x) = 0 \).
• Then, evaluate the second derivative \( f''(x) \) at each critical point.
• If \( f''(c) > 0 \), the function has a relative minimum at \( x=c \).
• If \( f''(c) < 0 \), the function has a relative maximum at \( x=c \).
• If \( f''(c) = 0 \), the test is inconclusive, and other tests like the first derivative test might be needed.

The intuition behind the test is related to the function's concavity:- Upward concavity at a critical point suggests a minimum, while downward concavity suggests a maximum.- The test's simplicity makes it a preferred method for quickly assessing critical points, especially when dealing with polynomial and differentiable functions.