Let f(x) =3 sin⁴x +10 sin ³ x +6 sin^2(x) -3, x belongs to [-pi/6,pi/2] then f is:- (a) increasing in (-pi/6,pi/2) b) decreasing in (0,pi/2) c) increasing in (-pi/6,0) d) decreasing in (-pi/6,0)

Key Ideas Needed

1. Derivative & Monotonicity
   • A function $f(x)$ is increasing where $f'(x) > 0$ and decreasing where $f'(x) < 0$.
2. Chain Rule for composite functions.
3. Trigonometric Bounds in the interval $x \in [-\pi/6, \pi/2]$:
   $\sin x \in [-0.5,\,1]\quad\text{and}\quad \cos x > 0\;\text{(everywhere except the endpoint }x = \pi/2).$

Logical Chain a Student Should Follow

1. Rewrite the problem purely in terms of $\sin x$.
2. Differentiate using the chain rule:
   • Write $t = \sin x$.
   • Treat $f$ as a polynomial in $t$.
   • Differentiate $f(t)$ with respect to $t$, then multiply by $\dfrac{dt}{dx} = \cos x$.
3. Factorise the derivative to make sign–checking easy.
4. Analyse the sign of every factor on the given interval:
   • $\cos x$ is positive.
   • The quadratic factor is positive (or zero only at $t = -0.5$).
   • The $\sin x$ factor decides the final sign (negative before $0$, positive after $0$).
5. Match the findings with the options.

Why each step?
Step 1 reduces the trigonometry mess to a neat polynomial.
Step 2 finds the slope.
Step 3 splits the slope into easy pieces.
Step 4 quickly tells us where the slope is + or −.
Step 5 finalises the answer.

🧒 ELI5 Version

Imagine you are riding a roller–coaster whose height is decided by the formula
$\text{height} = 3\,\sin^4x + 10\,\sin^3x + 6\,\sin^2x - 3$
Here, $x$ is like the position on the track measured in radians (a special angle–unit).
We want to know where the coaster goes up (increasing) and where it goes down (decreasing) between $-\pi/6$ and $\pi/2$.

How do we know that?

• We look at the slope of the track, called the derivative.
• If the slope is positive, the coaster is climbing (increasing).
• If the slope is negative, the coaster is going down (decreasing).

After checking, we find the slope is negative from $-\pi/6$ to $0$ and positive from $0$ to $\pi/2$.
So the only correct statement in the options is: The track is going down between $-\pi/6$ and $0$.

Step-by-Step Solution

1. Set substitution
   $t = \sin x \quad \Longrightarrow \quad f(x)=3t^4+10t^3+6t^2-3$

2. Differentiate wrt $x$
   First derivative wrt $t$:
   $\frac{df}{dt}=12t^3 + 30t^2 + 12t = 6t\,(2t^2 + 5t + 2)$
   Now apply the chain rule ($dt/dx=\cos x$):
   $f'(x)=\frac{df}{dt}\cdot\frac{dt}{dx}=6\sin x\,(2\sin^2x + 5\sin x + 2)\,\cos x$

3. Sign analysis of each factor

   • Factor 1: $\cos x > 0$ for $x \in (-\pi/2,\pi/2)$ (all our open interval).
   • Factor 2: $Q(\sin x)=2\sin^2x + 5\sin x + 2$
     Discriminant $\Delta = 5^2 - 4(2)(2)=25-16=9$.
     Roots: $\sin x = \frac{-5 \pm 3}{4} \Longrightarrow -2,\,-0.5$
     In our range $\sin x \in [-0.5,1]$, so $Q(\sin x) \ge 0$ and is strictly positive for $\sin x > -0.5$.
   • Factor 3: $\sin x$ itself.

4. Combine signs

   • For $x \in (-\pi/6,0):\; \sin x$ is negative $\Rightarrow f'(x)<0$ ⟹ decreasing.
   • For $x \in (0,\pi/2):\; \sin x$ is positive $\Rightarrow f'(x)>0$ ⟹ increasing.

5. Match with options
   a) Increasing in $(-\pi/6,\pi/2)$ ✖ (false).
   b) Decreasing in $(0,\pi/2)$ ✖ (false).
   c) Increasing in $(-\pi/6,0)$ ✖ (false).
   d) Decreasing in $(-\pi/6,0)$ ✔ (true).

Final Answer: Option (d)