Question

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Answer 1

The piecewise function f(x) is continuous but not differentiable at x = 1.

Is a piecewise function continuous and differentiable?

In this problem we must check if a piecewise function is both continuous and differentiable. This kind of function is continuous at x = a if and only if the lateral limits of the function for x = a is the same and differentiable if and only if the lateral limits of the first derivative for x = a is the same.

Continuity test

Now we determine the two lateral limits of the piecewise function at x = 1:

f(1) = 3 · e¹ ⁻ ¹ + 4 = 3 + 4 = 7

f(1) = 1² + 6 = 7

The function is continuous for x = 1.

Differentiability test

And the two lateral limits of the first derivatives of the piecewise function at x = 1:

f'(x) = 3 · eˣ ⁻ ¹

f'(1) = 3 · e¹ ⁻ ¹ = 3

f'(x) = 2 · x

f'(1) = 2 · 1 = 2

Hence, the piecewise function f(x) is continuous but not differentiable at x = 1.

To learn more on continuous functions: brainly.com/question/21447009

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