Optimization of a Piecewise Function

00:00 5 marks

Consider the function f defined by:

For \(x \leq 2\): \(f(x) = 3x + 1\)
For \(x > 2\): \(f(x) = -x^2 + 6x – 5\)

(a) Determine the maximum value of f(x) on each region of its domain.

(b) Based on your results, identify the global optimum (maximum) of the function over all real numbers.

Part a

5 minutes 3 marks

Question

For the function f(x) defined above, evaluate the maximum value on each region:

– For \(x \leq 2\) (linear piece): Determine the maximum value by evaluating the function at the endpoint.
– For \(x > 2\) (quadratic piece): Determine the maximum value by finding the vertex of the quadratic function.

Hint Hint

Hint

For the first piece, use the fact that the linear function is increasing. For the second piece, compute the vertex using the formula \(x = -\frac{b}{2a}\).

Answer Answer
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Answer Answer

Complete Answer

For \(x \leq 2\), the maximum value is 7 at \(x = 2\); for \(x > 2\), the maximum value is 4 at \(x = 3\).

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Part b

5 minutes 2 marks

Question

Compare the maximum values obtained in each region and state the global optimum of f(x) over all real numbers.

Hint Hint

Hint

Compare the maximum values found in part (a) to determine the overall maximum value of the function.

Answer Answer
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Explanation Explanation
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Answer Answer

Complete Answer

The global optimum is 7, occurring at \(x = 2\).

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