Question

find the critical point and determine if the function is increasing or decreasing on the given intervals. y=-x^2 + 4x + 5 critical point: c = the function is: on (-inf,c). on (c,inf).

Explanation:

Step1: Find the derivative

The derivative of $y = -x^{2}+4x + 5$ using the power - rule $(x^n)'=nx^{n - 1}$ is $y'=-2x + 4$.

Step2: Set the derivative equal to zero

Set $y' = 0$, so $-2x+4 = 0$. Solving for $x$ gives $2x=4$, and $x = 2$. So the critical point $c = 2$.

Step3: Test the intervals

Choose a test - point in the interval $(-\infty,2)$, say $x = 1$. Then $y'(1)=-2\times1 + 4=2>0$. So the function is increasing on $(-\infty,2)$.
Choose a test - point in the interval $(2,\infty)$, say $x = 3$. Then $y'(3)=-2\times3 + 4=-2<0$. So the function is decreasing on $(2,\infty)$.

Answer:

Critical point: $c = 2$. The function is increasing on $(-\infty,2)$ and decreasing on $(2,\infty)$.