To determine if $f(x)$ is differentiable at $x = 3$, we need to check both the continuity and differentiability conditions.

Step 1: Continuity at $x = 3$

For $f(x)$ to be differentiable at $x = 3$, it must first be continuous there. This requires: $\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3)$ From the piecewise function:

• For $x < 3$, $f(x) = x^2 + x$
• For $x \geq 3$, $f(x) = 2x^2 - 5x + 9$

1. Left-hand limit ($x \to 3^-$): $\lim_{x \to 3^-} f(x) = 3^2 + 3 = 9 + 3 = 12$

2. Right-hand limit ($x \to 3^+$): $\lim_{x \to 3^+} f(x) = 2(3^2) - 5(3) + 9 = 2(9) - 15 + 9 = 18 - 15 + 9 = 12$

3. Value of the function at $x = 3$: $f(3) = 2(3^2) - 5(3) + 9 = 12$

Since $\lim_{x \to 3^-} f(x) = \lim_{x \to 3^+} f(x) = f(3) = 12$, the function is continuous at $x = 3$.

Step 2: Differentiability at $x = 3$

Next, we need to check the left-hand and right-hand derivatives at $x = 3$.

1. Left-hand derivative ($x < 3$): The derivative of $f(x) = x^2 + x$ is: $f'(x) = 2x + 1$ So, the left-hand derivative at $x = 3$ is: $f'(3^-) = 2(3) + 1 = 6 + 1 = 7$

2. Right-hand derivative ($x \geq 3$): The derivative of $f(x) = 2x^2 - 5x + 9$ is: $f'(x) = 4x - 5$ So, the right-hand derivative at $x = 3$ is: $f'(3^+) = 4(3) - 5 = 12 - 5 = 7$

Since the left-hand and right-hand derivatives are equal, $f'(3^-) = f'(3^+) = 7$, the function is differentiable at $x = 3$, and the derivative is $f'(3) = 7$.

Final Answer:

$f'(3) = 7$

Would you like more details or have any further questions? Here are five related questions for deeper understanding:

1. What does it mean for a function to be continuous at a point?
2. How do you compute the derivative of a piecewise function?
3. What is the geometric interpretation of differentiability at a point?
4. Can a function be continuous but not differentiable? If so, can you provide an example?
5. How would the process change if the function was not continuous at $x = 3$?

Tip: When checking differentiability of piecewise functions, always verify continuity first, as differentiability implies continuity.