The length of the longest interval in which the function 3 sin x - 4 sin³ x is increasing, is (A) π/

Q: Why can't I just differentiate $3 \sin x - 4 \sin^3 x$ directly?

You absolutely can! If you differentiate it directly, you get $3 \cos x - 12 \sin^2 x \cos x$. You can then factor out $3 \cos x$ to get $3 \cos x (1 - 4 \sin^2 x)$. While this is correct, it is much harder to analyze for the increasing interval compared to the simplified form $\sin(3x)$. In JEE, recognizing the identity saves time and reduces the chance of calculation errors.

Q: Why is the answer $\frac{\pi}{3}$? Shouldn't the interval be larger?

The function $\sin(3x)$ is periodic with a period of $\frac{2\pi}{3}$. The question asks for the length of the *longest continuous interval* where the function is increasing. While the function increases infinitely many times across its domain, we are looking for the length of one such continuous segment. The segment $[-\frac{\pi}{6}, \frac{\pi}{6}]$ has a length of $\frac{\pi}{6} - (-\frac{\pi}{6}) = \frac{\pi}{3}$, which is the correct interval length.

Q: Do I need to worry about the general solution with $2n\pi$?

Great question! Since we are calculating the *length* of the interval, the specific location of the interval on the x-axis does not matter. Whether you look at the interval $[-\frac{\pi}{6}, \frac{\pi}{6}]$ or any other shifted interval where the function is increasing, the width remains the same. Focusing on the principal interval is sufficient to find the length.

Question

The length of the longest interval in which the function $3 \sin x - 4 \sin^3 x$ is increasing, is

Options:

(A) $\frac{\pi}{3}$

(B) $\frac{\pi}{2}$

(C) $\frac{3\pi}{2}$

(D) $\pi$

Sigma Percentile · Accepted Answer

### The Art of Unmasking: Simplifying the Function

Imagine you are standing before a complex, intimidating expression: $f(x) = 3 \sin x - 4 \sin^3 x$. At first glance, it looks like a mess of powers and trigonometric terms.

In the world of JEE mathematics, complexity is often a mask for elegance. This expression is the classic triple angle identity for sine:

$$\sin 3x = 3 \sin x - 4 \sin^3 x$$

By simply recognizing this, we transform our function into the much friendlier $f(x) = \sin 3x$. This is the first step in our journey—the art of simplification.

### The Calculus of Motion

Now that we have $f(x) = \sin 3x$, we need to determine where this function is increasing. As you know, the heartbeat of a function's growth is its derivative.

For a function to be increasing, its first derivative must be non-negative, meaning $f'(x) \ge 0$. Let's calculate that derivative.

Differentiating $\sin 3x$ with respect to $x$ requires the chain rule. The derivative of $\sin(u)$ is $\cos(u)$, and the derivative of $3x$ is $3$. Thus, we get:

$$f'(x) = 3 \cos 3x$$

It is elegant, isn't it? The complexity has vanished, leaving us with a clean, manageable expression.

### Solving the Inequality

We are now tasked with solving the inequality $3 \cos 3x \ge 0$. Since $3$ is a positive constant, we can divide both sides by $3$ without flipping the inequality sign, leaving us with $\cos 3x \ge 0$.

Now, visualize the unit circle. Where is the cosine function non-negative? It is non-negative in the first and fourth quadrants.

This means the angle $3x$ must lie within the interval:

$$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$$

This is the core geometric reality of the problem. We are looking for the longest continuous interval, and this range around the origin is exactly what we need.

### The Final Stretch

We have the inequality:

$$-\frac{\pi}{2} \le 3x \le \frac{\pi}{2}$$

To find the interval for $x$, we simply divide the entire inequality by $3$. This gives us:

$$-\frac{\pi}{6} \le x \le \frac{\pi}{6}$$

We have successfully isolated $x$. The final step is to calculate the length of this interval. The length of any interval $[a, b]$ is simply $b - a$.

Here, our interval is $[-\frac{\pi}{6}, \frac{\pi}{6}]$. So, the length is:

$$\frac{\pi}{6} - \left( -\frac{\pi}{6} \right) = \frac{\pi}{6} + \frac{\pi}{6} = \frac{2\pi}{6}$$

Simplifying this fraction, we arrive at our final answer: **$\frac{\pi}{3}$**. You have navigated the identity, mastered the derivative, and solved the inequality.

Animated Solution for Mathematics - Differentiation: The length of the longest interval in which the function $3 sin x - 4 sin^{3} x$ is increasing, is

Select Answer:

Visualized Solution

Identify the Function $f (x)$

Trigonometric Identity

Simplified Function

Condition for Increasing Function

Calculate the Derivative $f^{'} (x)$

Set Up the Inequality

Simplify the Inequality

Interval for Cosine

Find the Interval for $x$

Calculate the Interval Length

The Sigma Insight: Monotonicity

The Art of Unmasking

The Calculus of Motion

Solving the Inequality

The Final Stretch

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Animated Solution for Mathematics - Differentiation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The length of the longest interval in which the function 3sinx−4sin3x is increasing, is