Question

#Quality Question
@ Continuity and Diffrentiability

Let f(x) = (x+|x|)|x|
Then for all x

1) f is not continuous
2)f' is differentiable for all x
3) f' is continuous
4)none of the above ​

Answer 1

Answer:

$\sf \to f(x) = (x+|x|)|x|$

$\sf \to f(x) = x|x|+x^2$

For positive values, |x| = x,

For negative values, |x|= -x

$\to \displaystyle \sf \left \{ {{x > 0, \: f(x) = x^2+x^2} \atop {x < 0, \: f(x) = -x^2+x^2}} \right.$

$\to \displaystyle \sf \left \{ {{x > 0, \: f(x) = 2x^2} \atop {x < 0, \: f(x) = 0}} \right.$

f(x) = y = 2x² is a parabola opening upwards for x>0.

But for x<0, f(x) = y = 0 is the equation of the x-axis.

No sharp turns, no point discontinuities, and left and right-hand limits match at (0,0).

It is continuous, so it is differentiable at all points.

Option 2 and 3 is correct.