Question

determine whether the functions has maximum or minimum value. state the maximum or minimum value.
y = -x^2+4x - 3

Explanation:

Step1: Differentiate the function

Given \(y = -x^{2}+4x - 3\), the derivative \(y'=-2x + 4\).

Step2: Find critical points

Set \(y' = 0\), so \(-2x+4 = 0\). Solving for \(x\) gives \(2x=4\), then \(x = 2\).

Step3: Determine the second - derivative

Differentiate \(y'\) to get the second - derivative \(y''=-2\). Since \(y''<0\), the function has a maximum at \(x = 2\).

Step4: Find the maximum value

Substitute \(x = 2\) into the original function \(y=-(2)^{2}+4\times2 - 3=-4 + 8-3=1\).

Answer:

Maximum: yes, (2,1); Minimum: no, No value