Q.18. It is given that

$$f(x) = x^2 (x - 1)^2 (x - 2)$$ $$g(x) = -p(x - q)^2 (x - r)^2$$

where $p$, $q$, and $r$ are positive and $q < r$.

Find the set of values of $q$ and $r$ that guarantees the greatest number of distinct real solutions of the equation $f(x) = g(x)$ for all $p$.

Answer Choices

• $q < 1 \text{ and } r < 1$
• $q < 1 \text{ and } 1 < r < 2$
• $q < 1 \text{ and } r > 2$
• $1 < q < 2 \text{ and } 1 < r < 2$
• $1 < q < 2 \text{ and } r > 2$
• $q > 2 \text{ and } r > 2$

Solution

The correct answer is option 2: $q < 1 \text{ and } 1 < r < 2$.