Question

If [t] denotes the greatest integer ≤ t, then the number of points, at which the function
$f(x) = 4|2x + 3| + 9\lfloor x + \frac{1}{2} \rfloor - 12\lfloor x + 20 \rfloor$
is not differentiable in the open interval (–20, 20), is ____ .

Answer 1

$f(x) = 4|2x + 3| + 9\left\lfloor x + \frac{1}{2} \right\rfloor - 12\left\lfloor x + 20 \right\rfloor$
$=4|2x+3|+9[x+\frac{1}{2}]−12[x]−240$
f(x) is non differentiable at x $= \frac{-3}{2}$
and f(x) is discontinuous at {–19, –18, ….., 18, 19} as well as
\left\\{ -\frac{39}{2}, -\frac{37}{2}, \ldots, -\frac{3}{2}, -\frac{1}{2}, \frac{1}{2}, \ldots, \frac{39}{2} \right\\}
At same point, they are also non differentiable
∴ Total number of points of non differentiability
= 39 + 40
= 79
So, the correct answer is 79.