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Maths Question 16 – JEE-MAIN 2025

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Let f:R→R be a function defined by f(x)=||x+2|−2|. If m is the number of points of local minima and n is the number of points of local maxima of f, then m+n is

The graph of y=|g(x)| can be obtained by reflecting the part of y=g(x) that lies below the x-axis about the x-axis.

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Ninja Strategy•Graphical Transformation Insight

Recognize that f(x)=||x+2|−2| is a series of absolute value transformations. The graph of y=|g(x)| introduces minima at g(x)=0 and converts minima of g(x) (if g(x)<0) into maxima of |g(x)|. This structure typically leads to a 'W' or 'M' shape with a predictable number of extrema.

Step 1: Deconstruct the Absolute Value Function✦ Active

The function is f(x)=||x+2|−2|. To analyze its local extrema, we first define it piecewise by considering the critical points where the arguments of the absolute values become zero. The inner absolute value |x+2| changes definition at x=−2. The outer absolute value ||x+2|−2| changes definition where |x+2|−2=0, which implies |x+2|=2. This leads to x+2=2⟹x=0 or x+2=−2⟹x=−4. Thus, the critical points are x=−4,x=−2,x=0.

Step 2: Define f(x) Piecewise and Analyze Slopes○ Expand

We define f(x) in intervals based on the critical points:

f(x)={−x−4if x<−4(slope −1)x+4if −4≤x<−2(slope 1)−xif −2≤x<0(slope −1)xif x≥0(slope 1)
💡 Teacher's Secret Hint

Carefully handle the signs when removing absolute values in each interval.

Step 3: Identify Local Minima and Maxima○ Expand

By observing the changes in slope at the critical points:

- At x=−4: The slope changes from −1 to 1. This indicates a **local minimum**.

- At x=−2: The slope changes from 1 to −1. This indicates a **local maximum**.

- At x=0: The slope changes from −1 to 1. This indicates a **local minimum**.

Thus, there are m=2 local minima (at x=−4 and x=0) and n=1 local maximum (at x=−2). The total number of points of local extrema is m+n=2+1=3.

💡 Teacher's Secret Hint

A change in slope from negative to positive indicates a minimum, while positive to negative indicates a maximum.

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