For which values of a is f continuous at zero For which valu

For which values of a is f continuous at zero? For which values of a is f differentiable at zero? In this case, is the derivative function continuous? For which values of a is f twice differentiable?

Solution

For which values of a is f continuous at zero?

f(x) = x^a ; if x >= 0
f(x) = 0 if x < 0

The first function above can equal 0 only when \'a\' is a positive value

So, for a > 0, the function is continuous at 0 ---> FIRST ANSWER
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For which values of a is f differentiable at zero?

f(x) = x^a ; if x >= 0
f(x) = 0 if x < 0

f\'(x) = a*x^(a-1) ; if x >= 0
f\'(x) = 0 if x < 0

a*x^(a-1) will equal 0 only for a > 1

So, for a > 1, the function is differentiable at 0 ---> SECOND ANSWER
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In this case, is the derivative function continuous?

Since we do not have knowledge on the value of a, we cannot say for sure if it is contiuous. It would be continuous for a > 0
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For which values of a is f twice differentiable?

f\'\'(x) = a(a-1)*x^(a-2) ; x >= 0
f\'\'(x) = 0; x < 0

The above is differentiable at x = 0 only when exponent, a - 2 > 0 ---> a > 2

So, for all a > 2, it is twice differentiable ---> FOURTH ANSWER