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: https://brainly.com/question/21447009
#SPJ1
