Let f(x) be a function defined as `f(x)={{:(,int_(0)^(x)(3+|t-2|),"if "x gt 4),(,2x+8,"if "x le 4):}` Then, f(x) isA. continuous at x=4B. neither continuous nor differentiable at x=4C. everywhere continuous but not differentiable at x=4D. everywhere continuous and differentiable

Let f(x) be a function defined as `f(x)={{:(,int_(0)^(x)(3+|t-2|),"if

1.	Let f(x) be a function defined as `f(x)={{:(,int_(0)^(x)(3+\|t-2\|),"if "x gt 4),(,2x+8,"if "x le 4):}` Then, f(x) isA. continuous at x=4B. neither continuous nor differentiable at x=4C. everywhere continuous but not differentiable at x=4D. everywhere continuous and differentiable
Answer» Correct Answer - C For `x gt 4`, we have `f(x)=underset(0)overset(x)int (3+\|t-1\|)dt` `Rightarrow f(x)=underset(0)overset(2)int (3-\|t-2\|)dt+underset(0)overset(x)int (3+(t-2))dt` `Rightarrow f(x)=underset(0)overset(2)int (5-t)dt+underset(2)overset(x)int (1+t)dt` `Rightarrow f(x)=[5x-(t^(2))/(2)]^(2)+[t+(t^(2))/(2)]_(2)^(x)` `Rightarrow f(x)=(x^(2))/(2)+x+4` Thus, we have `f(x)={{:(,(x^(2))/(2)+x+4,"if "x gt 4),(,2x+8,"if "x le 4):}` Clearly, f(x) is continuous at x=4 We have `("LHD of f(x) at x=4")={(d)/(dx)(2x+8)}_(x=4)=2` `("RHD of f(x) at x=4")={(d)/(dx)(x^(2)/(2)+x+4)}_(x=4)=9` Clearly, `"Clearly", ("LHD of f(x) x=4") ne ("RHD of f(x) at x=4")` So, f(x) is not differentiable at x=4. Thus, f(x) is everywhere continuous but not differentiable at x=4.