Animated Solution for Mathematics - Differentiation: Let $f(x)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3,x\in [-\frac{\pi }{6},\frac{\pi }{2}]$. Then, $f$ is:

Welcome, fellow explorer of the mathematical universe. Today, we are not just solving a problem; we are embarking on a journey to understand the heartbeat of a function.

We have been given a function, $f(x)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3$, and we are tasked with understanding its behavior—where it climbs and where it falls—within the interval $[-\frac{\pi }{6},\frac{\pi }{2}]$.

When you first look at $f(x)$, it might seem intimidating. Powers of four, cubes, and squares can appear chaotic, but in the world of JEE Advanced, complexity is often a mask for elegance.

Notice that every term involves $\sin x$. This is a polynomial in disguise! If we let $u=\sin x$, our function becomes a simple polynomial in $u$. This perspective shift is the first step toward mastery.

To understand the monotonicity, we must find the derivative, $f^{\prime }(x)$. This represents the 'velocity' of our function.

Using the chain rule, where the derivative of ${\sin }^{n}x$ is $n{\sin }^{n-1}x\cos x$, we differentiate term by term:

Look at that expression; it is a symphony of trigonometric terms. Do not expand it further. Instead, look for the common rhythm.

We can factor out $6\sin x\cos x$ from every single term. This is the 'Aha!' moment. By pulling out the common factors, we reveal the hidden structure:

Now, we must factor the quadratic part, $2{\sin }^{2}x+5\sin x+2$. This is a classic quadratic factorization.

We split the middle term, $5\sin x$, into $4\sin x+\sin x$. This leads us to the fully factored form:

We now have four distinct factors: $6$, $\sin x$, $\cos x$, $(2\sin x+1)$, and $(\sin x+2)$. Our job is to determine the sign of each factor in the interval $[-\frac{\pi }{6},\frac{\pi }{2}]$.

1. The Constant: $6$ is always positive. 2. The Cosine: In the interval $[-\frac{\pi }{6},\frac{\pi }{2}]$, which spans the fourth and first quadrants, $\cos x$ is non-negative. 3. The Shifted Sine: $(\sin x+2)$ is always between $1$ and $3$, so it is strictly positive. 4. The Linear Factor: $(2\sin x+1)$. At $x=-\frac{\pi }{6}$, $\sin x=-\frac{1}{2}$, so $2(-0.5)+1=0$. For any $x>-\frac{\pi }{6}$, $\sin x>-\frac{1}{2}$, so $2\sin x+1>0$.

Do you see the beauty of this? Three of our factors are non-negative. The entire sign of $f^{\prime }(x)$ is dictated solely by $\sin x$.

When $\sin x<0$, the derivative is negative, and the function decreases. When $\sin x>0$, the derivative is positive, and the function increases.

In the interval $(-\frac{\pi }{6},0)$, $\sin x$ is negative, so the function is decreasing. In the interval $(0,\frac{\pi }{2})$, $\sin x$ is positive, so the function is increasing.

We have successfully mapped the terrain of this function. You have not just solved a problem; you have decoded the behavior of a mathematical object. Keep this analytical mindset, and no problem will ever be too complex for you.