If [t] denotes the greatest integer ≤ t , then number of points, at...

Given,

$f\left(x\right) = 4\left|2x + 3\right| + 9\left[x + \dfrac{1}{2}\right] - 12\left[x + 20\right]$

$x \in \left(- 20, 20\right)$

Now we know that \left[x\right]$is non-differentiable when$x \in I$</p><p>So,$12$\left[x + 20\right]$ is not differentiable at $x = I \in$\left\{- 19, - 18, \ldots 0, \ldots 19\right\} = 39$</p><p>Now$9\left[x + \dfrac{1}{2}\right]$is not differentiable when$x = I + \dfrac{1}{2}$</p><p>So, here also total non-differentiable points are$39$</p><p>Now checking modulus function</p><p>We know that modulus function are non-differentiable at point where modulus function is zero,</p><p>So at$x = \dfrac{- 3}{2}$modulus function is zero</p><p>Also$f$\left(x\right)$ at $x = \dfrac{- 3}{2}$ is not continuous and non-differentiable,

So, number of point of non-differentiability$= 39 + 39 + 1 = 79$