Question

Draw y=|x-5|+10 1) is it continuous at all points on the real line 2) is it diferentiable at all pint on the real line

Answer 1

it is given that a modulus function, y = |x - 5| + 10

To check : 1. it is continuous at all points on the real line.

2. it is differentiable at all points on the real line.

solution : let's brake the modulus function.

for x ≥ 5 , y = (x - 5) + 10 = x + 5

for x < 5 , y = -(x - 5) + 10 = -x + 15

we draw using above conditions we get graph as shown in figure.

here , Graph doesn't brake at any points. it means, function is continuous at all points on the real line.

we know in particular, a function is not differentiable at x = a if the graph of the function has a sharp point at that point.

you can see, graph has a sharp point at x = 5, so function is not differentiable at x = 5.

hence function is differentiable at all points on the real line except x = 5.