If the function
f(x)= 1/|x|, |x| ≥ 2 ; a x²+2 b, |x|<2
is differentiable on R, then 48(a+b) is equal to __________.

Solution

Maths sir · 4 steps

f is even, so check only x = 2: match the values, then match the slopes, and you get a and b.

1. Check one joint. Both pieces are even functions of x, so whatever works at x = 2 also works at x = -2. Near x = 2 the outer piece is 1/x, because |x| = x there.
2. Values must match. Differentiable means continuous first. At x = 2 the inner piece gives 4a + 2b and the outer piece gives 1/2, so 4a + 2b = 1/2.
3. Slopes must match. The inner piece has derivative 2ax, which is 4a at x = 2. The outer piece has derivative -1/x², which is -1/4 at x = 2.

   4a = -1/4 ⇒ a = -1/16, 2b = 1/2 + 1/4 = 3/4 ⇒ b = 3/8

4. So. a + b = -1/16 + 6/16 = 5/16, and 48 × 5/16 = 15.

The common slip: 9. Writing the derivative of 1/x as +1/x² gives a = 1/16 and b = 1/8, so the answer comes out as 9.

The idea to keep. At every joint of a piecewise function, differentiability gives two equations: the values match and the derivatives match. If the function is even, one joint is enough.

Answer from the NTA final answer key. Solution by PrepWiser.