Question

Let $\lambda = |x-2| + 2|x-4| + 4|x-6|$.

• A. If $\lambda > 8$ then above equation has 2 solutions
• B. If $\lambda < 10$ then above equation has no solutions
• C. If $\lambda = 8$ then above equation has 1 solutions
• D. If $\lambda > 6$ then above equation has more than 2 solutions

Answer 1

Answer: (A), (C)

Options (A) and (C) are correct.

Answer 2

Let $f(x) = |x-2| + 2|x-4| + 4|x-6|$.

We analyze $f(x)$ in different intervals:

• For $x < 2$: $f(x) = -(x-2) - 2(x-4) - 4(x-6) = -7x + 34$. $f(x)$ decreases from $\infty$ to $f(2)=20$.

• For $2 \le x < 4$: $f(x) = (x-2) - 2(x-4) - 4(x-6) = -5x + 30$. $f(x)$ decreases from $f(2)=20$ to $f(4)=10$.

• For $4 \le x < 6$: $f(x) = (x-2) + 2(x-4) - 4(x-6) = -x + 14$. $f(x)$ decreases from $f(4)=10$ to $f(6)=8$.

• For $x \ge 6$: $f(x) = (x-2) + 2(x-4) + 4(x-6) = 7x - 34$. $f(x)$ increases from $f(6)=8$ to $\infty$.

The minimum value of $f(x)$ is $8$ at $x=6$.

(A) If $\lambda > 8$, then $\lambda = f(x)$ will have two solutions (one $x<6$ and one $x>6$). This is Correct.

(B) If $\lambda < 10$, it has no solutions. This is incorrect, e.g., $\lambda=8$ has one solution.

(C) If $\lambda = 8$, then $\lambda = f(x)$ has one solution ($x=6$). This is Correct.

(D) If $\lambda > 6$, it has more than 2 solutions. This is incorrect; it has 2 solutions for $\lambda > 8$ and 1 solution for $\lambda=8$.