The function f(x) = sin⁴ x + cos⁴ x increases if (A) 0

Q: Why did we get $-\sin 4x$? Did I mess up the derivative?

You are doing great! The negative sign comes from the derivative of $\cos x$, which is $-\sin x$. It is a classic trap. Embrace the negative sign; it is part of the function's geometry. It tells us that the function is actually decreasing where the sine function is positive.

Q: Why is the answer $(\pi/4, 3\pi/8)$ when my calculation gave $(\pi/4, \pi/2)$?

Great observation. In JEE, sometimes the options provide a sub-interval. Since the function is increasing on the entire interval $(\pi/4, \pi/2)$, it must also be increasing on any subset of that interval, like $(\pi/4, 3\pi/8)$. Always check if your calculated interval contains the option provided.

Q: Can I simplify $f(x)$ using $1 - 2\sin^2 x \cos^2 x$ before differentiating?

Absolutely! That is a brilliant shortcut. It leads to $f(x) = 1 - \frac{1}{2}\sin^2 2x$. Differentiating this is often faster and less prone to chain-rule errors. Both paths lead to the same truth, but your intuition to simplify first is a hallmark of a top-tier problem solver.

Question

The function $f(x) = \sin^4 x + \cos^4 x$ increases if

Options:

(A) $0 < x < \pi/8$

(B) $\pi/4 < x < 3\pi/8$

(C) $3\pi/8 < x < 5\pi/8$

(D) $5\pi/8 < x < 3\pi/4$

Sigma Percentile · Accepted Answer

### The Rhythm of the Wave: Unlocking $f(x) = \sin^4 x + \cos^4 x$ Welcome, future engineer. Today, we are not just solving a calculus problem; we are peeling back the layers of a trigonometric wave. When you look at the function $f(x) = \sin^4 x + \cos^4 x$, do not see it as a static equation. See it as a heartbeat—a periodic oscillation that rises and falls with the rhythm of the unit circle. Our mission is to find the exact moments when this heartbeat is rising. In the language of calculus, we are hunting for the interval where the function is strictly increasing. ### Phase 1: The Calculus Lens To determine if a function is increasing, we must look at its slope. If the slope is positive, the function is climbing. Mathematically, this means we need to find the first derivative, $f'(x)$, and solve the inequality $f'(x) > 0$. This is our compass. Without it, we are just guessing; with it, we are navigating with precision. ### Phase 2: The Chain Rule Battle Now, let us differentiate. We have $f(x) = \sin^4 x + \cos^4 x$. Applying the power rule and the chain rule, we get: $$f'(x) = 4\sin^3 x \cdot \cos x + 4\cos^3 x \cdot (-\sin x)$$ I know this looks messy, but take a breath. Notice the symmetry. We have $4\sin^3 x \cos x$ and $-4\cos^3 x \sin x$. Let us factor out the common term, $4\sin x \cos x$. This leaves us with: $$f'(x) = 4\sin x \cos x (\sin^2 x - \cos^2 x)$$ ### Phase 3: The Trigonometric Dance This is where the elegance of trigonometry shines. We know that $2\sin x \cos x = \sin 2x$. Furthermore, we know that $\cos^2 x - \sin^2 x = \cos 2x$. Our expression has $\sin^2 x - \cos^2 x$, which is simply $-\cos 2x$. Substituting these identities, our derivative collapses beautifully: $$f'(x) = 2(2\sin x \cos x) (-(\cos^2 x - \sin^2 x))$$ $$f'(x) = 2(\sin 2x)(-\cos 2x) = -2\sin 2x \cos 2x$$ And with one final identity, $2\sin 2x \cos 2x = \sin 4x$, we arrive at the clean, powerful result: $$f'(x) = -\sin 4x$$ ### Phase 4: The Inequality Trap We need $f'(x) > 0$, which means $-\sin 4x > 0$. Here is the trap that catches many students: when you multiply by $-1$, you must flip the inequality sign. Thus, we are solving for $\sin 4x < 0$. Where is the sine function negative? In the third and fourth quadrants, which corresponds to the interval $(\pi, 2\pi)$. So, we set our argument $4x$ within these bounds: $$\pi < 4x < 2\pi$$ Dividing by $4$, we find the interval of increase: $$\frac{\pi}{4} < x < \frac{\pi}{2}$$ ### The Conclusion Our calculated interval is $(\frac{\pi}{4}, \frac{\pi}{2})$. Looking at our options, we see $(\frac{\pi}{4}, \frac{3\pi}{8})$. Since $\frac{3\pi}{8}$ is less than $\frac{\pi}{2}$, this interval is a subset of our solution. The function is indeed increasing here. You have successfully navigated the calculus, the trigonometry, and the inequality logic. Keep this confidence—you are ready for the next challenge.

Animated Solution for Mathematics - Differentiation: The function $f (x) = sin^{4} x + cos^{4} x$ increases if

Select Answer:

Visualized Solution

Identify the Function $f (x)$

Condition for Increasing Function

Differentiate $f (x)$

Apply the Chain Rule

Factor the Derivative

Apply Double Angle Identities

Simplify to Final Derivative

Set Up the Inequality

Solve for the Angle $4 x$

Find the Interval for $x$

Match with the Given Options

The Sigma Insight: Monotonicity

The Rhythm of the Wave

Phase 1

Phase 2

Phase 3

Phase 4

The Conclusion

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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 function f(x)=sin4x+cos4x increases if