Animated Solution for Mathematics - Differentiation: If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x)=9x^4+12x^3-36x^2+25,x\in R$, then :

Imagine you are standing at the base of a massive, undulating roller coaster track. This track is defined by the function $f(x)=9x^4+12x^3-36x^2+25$.

As you look at this fourth-degree polynomial, you might feel a sense of intimidation. However, in the realm of JEE Advanced, we visualize the geometry of the universe to map the topography of the function by identifying its local minima (valleys) and local maxima (peaks).

To find these points, we must determine where the roller coaster levels out. Mathematically, this occurs where the slope of the tangent line is zero. We invoke the First Derivative Test:

Applying the power rule, we obtain:

Setting this derivative to zero, we factor out $36x$:

This simplifies to the factored form:

The critical points are $x=-2$, $x=0$, and $x=1$. These represent the 'turning points' of our journey.

To distinguish between peaks and valleys, we use the Second Derivative Test. We compute the second derivative to determine the curvature:

If the curvature is positive, the curve is 'cupping' upwards (a bowl), indicating a minimum. If it is negative, the curve is 'frowning' downwards (a hill), indicating a maximum.

For $x=-2$:

Since $216>0$, the curve is concave up, confirming that $x=-2$ is a local minimum.

For $x=0$:

Since $-72<0$, the curve is concave down, confirming that $x=0$ is a local maximum.

For $x=1$:

Since $108>0$, the curve is concave up, confirming that $x=1$ is a local minimum.

We have systematically dissected the function. Our local minima occur at $x=-2$ and $x=1$, forming the set $S_1=\{-2,1\}$.

Our only local maximum occurs at $x=0$, forming the set $S_2=\{0\}$. By applying this systematic approach, you have successfully navigated the roller coaster of calculus.