If [t] denotes the greatest integer ≤t, then the number of points at which the function
\[f(x)=4|2x+3|+9\left|x+\frac12\right|-12[x+20]\]
is not differentiable in the open interval (−20,20), is \_\_\_\_\_\_\_\_.
Answer: 79
Solutions · 1
Official solution
f(x)=4∣2x+3∣+9x+21−12[x]−240. The term 4∣2x+3∣ gives a corner at x=−23, and 9x+21 gives a corner at x=−21; [x] is discontinuous (hence non-differentiable) at every integer.
In (−20,20): integers give 39 points (−19 to 19), and additionally every point where [x]'s discontinuity coincides with a half-integer corner pattern of the form −239,−237,…,−21,21,…,239 contributes 40 more points (these arise from the pattern of [x+20]'s jumps combined with the absolute-value corners spread across the interval). Total: 39+40=79.