Consider a function f defined on R by f (x) = (sin2 πx | x2 – 3x + 2 | + |3x – 4|) [x2 – 1], then which of the following hold(s) true? [Note: [k] denotes greatest integer function less than or equal to k.]

Engineering

Mathematics

Important Points on Derivability

Question

Consider a function f defined on R by f (x) = (sin² πx | x² – 3x + 2 | + |3x – 4|) [x² – 1], then which of the following hold(s) true?
[Note: [k] denotes greatest integer function less than or equal to k.]

there are 7 points of discontinuity of f(x) in [–2, 2].

there are 7 points of non differentiability in (–2, 2).

there are 8 points of discontinuity of f(x) in [–2, 2].

there are 6 points of non differentiability in (–2, 2).

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$f (x) = (\underset{g (x)}{\underset{⏟}{\sin^{2} πx | x^{2} - 3 x + 2 | + | 3 x}} - 4 |) [x^{2} - 1]$

Clearly g is continuous on [–2, 2] and g(x) $\neq$ 0 on [–2, 2]
⇒     f is discontinuous at x² – 1 = 1, 2, 3, 0
       x² = 1, 2, 3, 4
       x = ±1, ± $\sqrt{2}, \pm \sqrt{3}$ , ±2 i.e. at 8 points
Again     f (x) = g(x) [x² – 1]
       g is non differentiable at only x = $\frac{4}{3}$
but $\frac{4}{3}$ x = is not a point of non differentiable for f (x) because $[{(\frac{4}{3})}^{2} - 1]$ = 0
while [x² – 1] is non differentiable at x² – 1 = 0, 1, 2 ⇒ x = ±1, $\pm \sqrt{2}, \pm \sqrt{3}$
therefore f is non derivable at 6 points.