Answer:
A)
B) Continuous but not differentiable.
Step-by-step explanation:
So we have the piecewise function:
A)
To write the differentiated piecewise function, let's differentiate each equation separately. Thus:
1)
Expand:
The derivative of a linear equation is just the slope. The derivative of a constant is 0. Thus:
2)
Again, the derivative of a constant is 0. Thus:
3)
We have:
Expand:
Simplify:
Now, let's substitute our original equations for the differentiated equations. The inequalities will stay the same. Therefore:
B)
For a function to be differentiable at a point, the function <em>must </em>be a) continuous at that point, and b) the left and right hand derivatives must be equivalent.
Let's first determine if the function is continuous at the point. Remember that a function is continuous at a point if and only if:
Let's find the left hand limit of f(x) at it approaches 2.
Since it's coming from the left, let's use the third equation:
Direct substitution:
So:
Now, let's find the right-hand limit:
Since we're coming from the right, let's use the first equation:
Direct substitution:
Multiply and add:
So, both the left and right hand limits are equivalent. Now, find the limit at x=2.
From the piecewise function, we can see that the value of f(2) is 6.
Therefore, the function is continuous at x=2.
Now, let's determine differentiability at x=2.
For a function to be differentiable at a point, both the right hand and left hand derivatives must be equivalent.
So, let's find the derivative of the function as x approahces 2 from the left and from the right.
From the differentiated piecewise function, we can see that as x approaches 2 from the left, the derivative is 1.
As x approaches 2 from the right, the derivative is 1/2.
Therefore, the right and left hand derivatives are <em>not</em> the same.
Thus, the function is continuous but <em>not</em> differentiable.