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QJEEM22Mab605: JEE Main 2022 Mathematics - Class 12 JEEM22Mab605 Solution: Let where denotes the greatest integer less than or equal to .…

Let f(x)=\left\{\begin{array}{l}\left|4 x^{2}-8 x+5\right|, \text { if } 8 x^{2}-6 x+1 \geqslant 0 \\ {\left[4 x^{2}-8 x+5\right], \text { if } 8 x^{2}-6 x+1<0,}\end{array}\right. where [α][\alpha] denotes the greatest integer less than or equal to α\alpha . Then the number of points in R\mathbf{R} where ff is not differentiable is ___________.

Step-by-Step Verified Solution

f(x)={∣4x2−8x+5∣,amp; if 8x2−6x+1≥0[4x2−8x+5],amp; if 8x2−6x+1lt;0f(x)= \begin{cases}\left|4 x^{2}-8 x+5\right|, & \text { if } 8 x^{2}-6 x+1 \geq 0 \\ {\left[4 x^{2}-8 x+5\right],} & \text { if } 8 x^{2}-6 x+1<0\end{cases}

={4x2−8x+5,amp; if x∈[−∞,14]∪[12,∞)[4x2−8x+5]amp; if x∈(14,12)= \begin{cases}4 x^{2}-8 x+5, & \text { if } x \in\left[-\infty, \frac{1}{4}\right] \cup\left[\frac{1}{2}, \infty\right) \\ {\left[4 x^{2}-8 x+5\right]} & \text { if } x \in\left(\frac{1}{4}, \frac{1}{2}\right)\end{cases}

f(x)={4x2−8x+5amp; if x∈(−∞,14]∪[12,∞)3amp;x∈(14,2−22)2amp;x∈[2−22,12)f(x)=\left\{\begin{array}{cc}4 x^2-8 x+5 & \text { if } x \in\left(-\infty, \frac{1}{4}\right] \cup\left[\frac{1}{2}, \infty\right) \\3 & x \in\left(\frac{1}{4}, \frac{2-\sqrt{2}}{2}\right) \\2 & x \in\left[\frac{2-\sqrt{2}}{2}, \frac{1}{2}\right)\end{array}\right.

JEE Main 2022 (Online) 25th July Morning Shift Mathematics - Limits, Continuity and Differentiability Question 76 English Explanation
∴\therefore \quad Non-diff at x=14,2−22,12x=\frac{1}{4}, \frac{2-\sqrt{2}}{2}, \frac{1}{2}

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