1
JEE Main 2022 (Online) 29th July Evening Shift
Numerical
+4
-1

If $$[t]$$ denotes the greatest integer $$\leq t$$, then the number of points, at which the function $$f(x)=4|2 x+3|+9\left[x+\frac{1}{2}\right]-12[x+20]$$ is not differentiable in the open interval $$(-20,20)$$, is __________.

2
JEE Main 2022 (Online) 28th July Morning Shift
Numerical
+4
-1

Let $$f:[0,1] \rightarrow \mathbf{R}$$ be a twice differentiable function in $$(0,1)$$ such that $$f(0)=3$$ and $$f(1)=5$$. If the line $$y=2 x+3$$ intersects the graph of $$f$$ at only two distinct points in $$(0,1)$$, then the least number of points $$x \in(0,1)$$, at which $$f^{\prime \prime}(x)=0$$, is ____________.

3
JEE Main 2022 (Online) 28th July Morning Shift
Numerical
+4
-1

$$\lim\limits_{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+3 \sin (x+2)}\right)^{\frac{100}{x}}$$ is equal to ___________.

4
JEE Main 2022 (Online) 25th July Morning Shift
Numerical
+4
-1

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 $$[\alpha]$$ denotes the greatest integer less than or equal to $$\alpha$$. Then the number of points in $$\mathbf{R}$$ where $$f$$ is not differentiable is ___________.