The function f(x) is continuous but not differentiable at x = 1.
How to check the continuity and differentiability of a piecewise function?
A piecewise function is continuous if and only if the lateral limits are the same and differentiable if and only if the lateral limits of the first derivatives are the same.
Continuity
Now we determine the lateral limits of the piecewise function at x = 1:
f(1) = 3 · e¹ ⁻ ¹ + 4 = 3 + 4 = 7
f(1) = 1² + 6 = 1 + 6 = 7
The piecewise function is continuous at x = 1.
Differentiability
And the lateral limits of the first derivatives of the piecewise function:
f'(x) = 3 · eˣ ⁻ ¹
f'(1) = 3 · e¹ ⁻ ¹ = 3
f'(x) = 2 · x
f'(1) = 2 · 1 = 2
The piecewise function is not differentiable at x = 1.
So the function f(x) is continuous but not differentiable at x = 1.
To learn more on piecewise functions: https://brainly.com/question/12561612
#SPJ1
