Question

find the critical point and determine if the function is increasing or decreasing on the given intervals.
y = -x^2 + 8x + 5
critical point: c =
the function is:
on (-∞, c).
on (c, ∞).

Explanation:

Step1: Find the derivative

Differentiate $y = -x^{2}+8x + 5$ using power - rule. The derivative $y'=-2x + 8$.

Step2: Find the critical point

Set $y' = 0$. So, $-2x+8 = 0$. Solving for $x$ gives $2x=8$, then $x = 4$. So the critical point $c = 4$.

Step3: Test the intervals

For the interval $(-\infty,4)$, choose a test - point, say $x = 3$. Then $y'(3)=-2\times3 + 8=2>0$. So the function is increasing on $(-\infty,4)$.
For the interval $(4,\infty)$, choose a test - point, say $x = 5$. Then $y'(5)=-2\times5 + 8=-2<0$. So the function is decreasing on $(4,\infty)$.

Answer:

Critical point: $c = 4$
The function is:
increasing on $(-\infty,4)$
decreasing on $(4,\infty)$