Q.1

If f(5+x) = f(5-x) for every real x, and f(x) = 0 has four distinct real roots, then what is the sum of these roots?

TITA type i.e. Type In The Answer type

1. Re-write the given relation in the form, f(x) = f(?)

2. If p is a root, can you find another root related to p? What are their sum?

20

f(5 + x) = f(5 – x)

Denoting (5 + x) = y,

we have x = y – 5.

And (5 – x) = 5 – (y – 5) i.e. 10 – y.

Replacing (5 + x) with y and (5 – x) with (10 – y),

f(y) = f(10 – y), for all values of y.

If p be a root of f(x)

⇒ f(p) = 0,

And using the earlier relation, f(10 – p) = 0 i.e. (10 – p) is also a root.

Similarly, if q is another root of f(x). Then (10 – q) is also a root.

Adding all 4 root, p + (10 – p) + q + (10 – q) = 20