Piecewise Function Continuity and Differentiability

00:00 6 marks

Consider the piecewise function defined by:

f(x) = { x^2, for x \(\leq\) 2;
m\,x + b, for x > 2 }.

Determine the values of m and b that ensure the function is both continuous and differentiable at x = 2.

Part a

5 minutes 3 marks

Question

Determine the condition for continuity of f(x) at x = 2 by equating the left-hand and right-hand limits. Express the resulting relationship between m and b.

Hint Hint

Hint

For continuity at x=2, ensure the function values from both pieces are equal at x=2. Substitute x=2 into both expressions and equate them.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

The continuity condition is \(2m + b = 4\) (or equivalently, \(b = 4 – 2m\)).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet

Part b

5 minutes 3 marks

Question

Determine the value of m required for differentiability at x = 2. Then, using the continuity condition from part (a), find the corresponding value of b.

Hint Hint

Hint

Differentiate each piece of f(x) and equate the derivatives at x = 2 to ensure the slopes match from both sides.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

Differentiability requires that the left-hand derivative, \(4\), equals the right-hand derivative, m. Therefore, m = 4. Substituting into the continuity condition gives b = \(4 – 2(4) = -4\).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet