Piecewise Function Slope Analysis

00:00 4 marks

Consider the function \( f(x) = \begin{cases} 2x + 1, & x < 3 \\ x^2 – 4, & x \geq 3 \end{cases} \).

(a) Find the derivative \( f'(x) \) for each interval.

(b) Determine whether \( f(x) \) is differentiable at \( x = 3 \).

Hint Hint

Hint

Differentiate each piece separately and check if the left-hand derivative and right-hand derivative are equal at \( x = 3 \). Also, remember to check continuity at \( x=3 \) as a necessary condition for differentiability.

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Answer Answer

Complete Answer

For \( x < 3 \), \( f'(x) = 2 \); for \( x \geq 3 \), \( f'(x) = 2x \). The left-hand derivative at \( x = 3 \) is 2 and the right-hand derivative is \( 2(3) = 6 \). Since the left-hand derivative is not equal to the right-hand derivative at \( x = 3 \), \( f(x) \) is not differentiable at \( x = 3 \).

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