Problem 74 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f\) be a continuous and differentiable function in \((a, b), \lim _{x \rightarrow a^{+}} f(x) \rightarrow \infty\) and \(\lim _{x \rightarrow b^{-}} f(x) \rightarrow-\infty .\) If \(f^{\prime}(x)\) \(+f^{2}(x) \geq-1\) for \(a

Step-by-Step Solution

Answer

The answer is (A) \(b-a \leq \pi\).

Step 1: Understand the given information

We have a continuous and differentiable function \( f \) on the interval \((a, b)\). The function satisfies \( \lim_{x \rightarrow a^+} f(x) = \infty \) and \( \lim_{x \rightarrow b^-} f(x) = -\infty \). We also know that \( f'(x) + f^2(x) \geq -1 \) for \( a < x < b \). We need to find a condition for the length of the interval \( b-a \).

Step 2: Apply inequalities related to the function

Consider the inequality:\[ f'(x) + f^2(x) \geq -1. \]By rearranging, we get:\[ f'(x) \geq -1 - f^2(x). \]

Step 3: Graphically analyze the behavior of the function

The condition \( f'(x) + f^2(x) \geq -1 \) suggests that \( f(x) \) decreases gradually from large positive values to large negative values. The derivative inequality controls how steep the decrease can be.

Step 4: Integrate the inequality over the interval (a, b)

To analyze \( b-a \), let's integrate the rearranged inequality from \( a \) to \( b \):\[ \int_a^b f'(x) \geq \int_a^b (-1 - f^2(x)) \, dx. \]The left side \( \int_a^b f'(x) \, dx = f(b) - f(a) = (-\infty) - (\infty) = - \infty. \)

Step 5: Evaluate the implications for b-a

Since the left side of the inequality evaluates to \(-\infty\), consider the integral of \(-1\) on the right:\[ - (b-a) - \int_a^b f^2(x) \, dx \]This implies that for the inequality to hold, the length of the interval \( b-a \) cannot be arbitrarily long, given the constraints on \( f(x) \). Using this kind of method often relates to the properties of specific functions, for instance, trigonometric functions over their periods.

Step 6: Consider functions with known behavior over intervals

Trigonometric functions like tangent and sine have periods where they transition from \( \infty \) to \(-\infty\) (or vice versa) and could match the described properties over intervals like \((0, \pi)\) for tangent. Thus, for such functions, \( b-a \) could be at most \( \pi \), suggesting a connection to option (A).

Step 7: Conclusion considering behavior

Based on the trigonometric interpretation and the constraints given, the total change (maximum reach from +infty to -infty) in such functions doesn't allow passing beyond a total interval length \( b-a \leq \pi \). Hence, the answer aligns with the inequality in option (A).

Key Concepts

Continuous Functions

In calculus, a continuous function is one that does not have any interruptions, jumps, or breaks in its domain. That means for any given point within the interval, the function will behave predictably.
While dealing with continuous functions, we often refer to the property that the limit of the function as it approaches any point from either side (left or right) must equal the value of the function at that point.

• If you have a function that is continuous over an interval like \((a, b)\), you won’t encounter any interpretations or sudden changes in value that aren't accounted by simply looking at nearby values.
• This behavior is crucial since we are assured that the extreme behavior at points, like infinity as it approaches from a side, is smoothly transitioned.
• For instance, in our problem, \(f(x)\) showed substantial behavior at the bounds of an interval, going from \infty\ to -\infty\ without any breaks in between.

In practical terms, when solving calculus problems involving these kinds of transitions, continuity ensures that we can smoothly apply methods like integration, significantly simplifying the calculation of integral limits.

Differentiable Functions

A differentiable function is a type of function that has a derivative, meaning it can be represented by a tangent line at every point within its domain. Differentiability implies continuity, but not every continuous function is differentiable.

• Differentiable functions possess derivatives across their intervals, allowing us to make statements about the slope or the rate of change of the function.
• In our exercise, knowing that \(f\) is differentiable allows us to use inequalities involving the derivative \(f'(x)\).
• For the inequality \(f'(x) + f^2(x) \geq -1\), it provides insights into the function's rising or falling trend.
• This condition ensures that although \(f\) is moving from a high positive value to a low negative one, its slope or rate of change is bounded in a certain way.

The power of differentiability in calculus lies in being able to predict and describe the dynamic behavior of functions, because understanding derivative behavior helps manage expectations on the nature and limits of their growth or decay.

Trigonometric Behavior in Calculus

Trigonometric functions are often used to explore limits, periodicity, and behavior of functions within particular intervals. Looking at our exercise, there is a potential parallel with trigonometric functions like sine and tangent, as they also exhibit behavior transitioning between \( \infty \) and \( -\infty \) over certain intervals.

• The tangent function, for instance, is well-known to have undefined points, which results in its graph having vertical asymptotes as it approaches \infty\ or -\infty\ within each of its periods (typically every \(rac{\pi}{2}\) along the x-axis).
• This behavior closely mirrors the behavior discussed in this problem scenario, tying in the nature of \(f(x)\) as it swings from positive to negative infinity over an interval.
• The insight here connects that trigonometric characteristics can severely restrict intervals like \([0, \pi]\) when describing the full behavior of these functions.

Applying knowledge from trigonometric function behavior in calculus allows for the creation of reasoning strategies that relate inequality conditions of derivatives back to real-world formulas that predict intervals limits.