Let f(x)=\(\left\{\begin{matrix} |4x^2-8x+5|, & if\,8x^2-6x+1\geq0 \\ |4x^2-8x+5|, & if\,8x^2-6x+1<0 \end{matrix}\right.\) where [α] denotes the greatest integer less than or equal to α.Then the number of points in R where f is not differentiable is
Correct Answer: 3
Solution and Explanation
f(x)=\(\left\{\begin{matrix} |4x^2-8x+5|, & if\,8x^2-6x+1\geq0 \\ |4x^2-8x+5|, & if\,8x^2-6x+1<0 \end{matrix}\right.\)
=\(\left\{\begin{matrix} |4x^2-8x+5|, & if\,x\in [-\infty,\frac{1}{4}]\bigcup[\frac{1}{2},\infty] \\ |4x^2-8x+5|, & if\, x\in(\frac{1}{4},\frac{1}{2})\end{matrix}\right.\)
f(x)=\(f(x)=\left\{\begin{matrix} 4x^2-8x+5 &if\,x\in [-\infty,\frac{1}{4}]\bigcup[\frac{1}{2},\infty] & \\ 3&x\in(\frac{1}{4},\frac{2-\sqrt2}{2}) & \\ 2& x\in(\frac{2-\sqrt2}{2},\frac{1}{2}) & \end{matrix}\right.\)![[α] denotes the greatest integer less than or equal to α](https://images.collegedunia.com/public/qa/images/content/2024_01_05/jeemain202_29b5ed061704459101021.png)
∴ Non-diff at
x=\(\frac{1}{4}\),\(\frac{2-\sqrt2}{2}\),\(\frac{1}{2}\)
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Concepts Used:
Addition of Vectors
A physical quantity, represented both in magnitude and direction can be called a vector.
For the supplemental purposes of these vectors, there are two laws that are as follows;
- Triangle law of vector addition
- Parallelogram law of vector addition
Properties of Vector Addition:
- Commutative in nature -
It means that if we have any two vectors a and b, then for them
\(\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{a}\)
- Associative in nature -
It means that if we have any three vectors namely a, b and c.
\((\overrightarrow{a}+\overrightarrow{b})+\overrightarrow{c}=\overrightarrow{a}+(\overrightarrow{b}+\overrightarrow{c})\)
- The Additive identity is another name for a zero vector in vector addition.
Read More: Addition of Vectors