Question

directions: for each of the following polynomial graphs, determine the global minimum and global maximum. if the graph does not have a specific global extrema, write
one\ in the appropriate space
global minimum = - 1 at x = 3
global maximum = 2 at x = 5
global minimum = - 1 at x = 1
global maximum = - 1 at x = 2
directions: for each of the following, determine if the given polynomial must have a global minimum, global maximum or neither. explain your reasoning.

1. f(x)=x^4 - 5x^2+x + 6
2. y=-2x^3 - x^2+8x
3. g(x)=-x^4+x^3+4x^2 - 1

i dont have de smos to do this
directions: for the following polynomial graphs, determine any x - values where the function has a point of inflection.

Explanation:

Step1: Analyze polynomial 15

For \(f(x)=x^{4}-5x^{2}+x + 6\), since the leading - coefficient of the even - degree polynomial (\(n = 4\) and \(a_4=1>0\)) is positive, as \(x\to\pm\infty\), \(y\to+\infty\). So, the function has a global minimum. It does not have a global maximum because the function values increase without bound as \(x\to\pm\infty\).

Step2: Analyze polynomial 16

For \(y=-2x^{3}-x^{2}+8x\), it is a cubic (odd - degree) polynomial. As \(x\to+\infty\), \(y\to-\infty\) (because the leading coefficient \(a_3=-2 < 0\)), and as \(x\to-\infty\), \(y\to+\infty\). So, it has neither a global maximum nor a global minimum.

Step3: Analyze polynomial 17

For \(g(x)=-x^{4}+x^{2}+4x - 1\), since it is an even - degree polynomial with a negative leading coefficient (\(n = 4\) and \(a_4=-1<0\)), as \(x\to\pm\infty\), \(y\to-\infty\). So, the function has a global maximum and does not have a global minimum because the function values decrease without bound as \(x\to\pm\infty\).

Answer:

1. Global minimum, no global maximum
2. Neither global maximum nor global minimum
3. Global maximum, no global minimum