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Given: f(x)= {1/x, if 0<x<b                   {1-1/(4x), if b less than or equal to x A. Determine a value of b so that f is continuous at b B. Is f differentiable at the value of b? Verify this using the definiton of derivative

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Given `f(x)= {[1/x,0<x<b],[1-1/(4x),b<=x]}.`

(a)  `1/x` is continuous for all values of x, x>0. Also `1-1/(4x)` is continuous for all values of x>0.

For f(x) to be continuous at b, we need `1/x` evaluated at b to be equal to `1-1/(4x)` evaluated at b.

`1/x=1-1/(4x) ==> 4=4x-1 ==> x=5/4`

f(x) is continuous at b if `b=5/4`

(b) To determine if f(x) is differentiable at `b=5/4` :












Since the left-hand limit does not agree with the right-hand limit, the function is not differentiable at `x=5/4`

The graph:

Note the "cusp" at `x=5/4`


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