Question

If
\[ f(x)=\max\{3-x,\;3+x,\;6\} \] is not differentiable at \(x=a\), and \(x=b\), then \(|a|+|b|=\)

• A. \(4\)
• B. \(5\)
• C. \(6\)
• D. \(8\)

Hint:

For functions involving \(\max\), non-differentiability usually occurs at points where the maximum switches from one expression to another.

Answer 1

The Correct Option is C

Step 1: Identify the three functions.
Given,
\[ f(x)=\max\{3-x,\;3+x,\;6\} \]
Let
\[ y_1=3-x,\qquad y_2=3+x,\qquad y_3=6 \]
The function \(f(x)\) will be non-differentiable at points where the maximum changes from one function to another.

Step 2: Find intersection points with \(y=6\).
First compare
\[ 3-x=6 \]
\[ -x=3 \]
\[ x=-3 \]
Now compare
\[ 3+x=6 \]
\[ x=3 \]
So, possible non-differentiable points are
\[ x=-3,\quad x=3 \]

Step 3: Check which function is maximum in intervals.
For \(x\lt -3\), \(3-x\gt 6\), so
\[ f(x)=3-x \]
For \(-3\lt x\lt 3\), both \(3-x\lt 6\) and \(3+x\lt 6\), so
\[ f(x)=6 \]
For \(x\gt 3\), \(3+x\gt 6\), so
\[ f(x)=3+x \]
Thus, \(f(x)\) changes expression at
\[ x=-3 \] and
\[ x=3 \]
Hence,
\[ a=-3,\qquad b=3 \]

Step 4: Find \(|a|+|b|\).
\[ |a|+|b|=|-3|+|3| \]
\[ =3+3 \]
\[ =6 \]

Step 5: Final conclusion.
Hence,
\[ \boxed{6} \]