Given that y=2x62+px+16 and that y<0 only when 2<x<k, find the value of p and k
To find the value of p and k, we need to use the given information that y<0 when 2<x<k.
Let's start by solving the given equation: y = 2x^2 + px + 16.
Since we know that y<0 when 2<x<k, we can substitute those values into the equation to form two inequalities:
When x = 2:
y = 2(2)^2 + p(2) + 16
y = 8 + 2p + 16
y = 2p + 24 (equation 1)
We also know that y<0 when x = 2, so:
2p + 24 < 0 (inequality 1)
When x = k:
y = 2(k)^2 + p(k) + 16
y = 2k^2 + pk + 16 (equation 2)
We also know that y<0 when x = k, so:
2k^2 + pk + 16 < 0 (inequality 2)
Now, we need to find the values of p and k that satisfy both inequality 1 and inequality 2.
Let's solve inequality 1 first:
2p + 24 < 0
2p < -24
p < -12
Next, let's solve inequality 2:
2k^2 + pk + 16 < 0
Unfortunately, we cannot solve this inequality directly because we have two variables, k and p. We need more information to solve it.
Therefore, with the given information, we can find the value of p as p < -12. However, we cannot determine the value of k without additional constraints or information.