Question

the polynomial function f is given by ( f(x) = ax^4 + bx^3 + cx^2 + dx + k ), where ( a
eq 0 ) and ( b, c, d, ) and ( k ) are constants. which of the following statements about ( f ) is true? a ( f ) has both a global maximum and a global minimum. b ( f ) has either a global maximum or a global minimum, but not both. c ( f ) has neither a global maximum nor a global minimum. d the nature of a global maximum or a global minimum for ( f ) cannot be determined without more information about ( b, c, d, ) and ( k ).

Explanation:

Step1: Analyze the degree and leading coefficient

The polynomial \( f(x) = ax^4 + bx^3 + cx^2 + dx + k \) is a fourth - degree polynomial (since the highest power of \( x \) is 4) with \( a
eq0 \). For a polynomial function of the form \( y = a_nx^n+\cdots+a_1x + a_0 \), as \( x
ightarrow\pm\infty \), the behavior of the function is determined by the leading term \( a_nx^n \). Here, the leading term is \( ax^4 \). Since the degree \( n = 4 \) (even) and \( a
eq0 \):

• If \( a>0 \), as \( x

ightarrow\infty \), \( ax^4
ightarrow\infty \) and as \( x
ightarrow-\infty \), \( ax^4
ightarrow\infty \) (because raising a negative number to an even power gives a positive result). So the function will have a global minimum (because the graph opens upwards and the function values go to \( \infty \) as \( x \) approaches \( \pm\infty \), so there must be a lowest point on the graph).

• If \( a < 0 \), as \( x

ightarrow\infty \), \( ax^4
ightarrow-\infty \) and as \( x
ightarrow-\infty \), \( ax^4
ightarrow-\infty \) (because a negative number times a positive number (from \( x^4 \)) is negative). So the function will have a global maximum (because the graph opens downwards and the function values go to \( -\infty \) as \( x \) approaches \( \pm\infty \), so there must be a highest point on the graph).

Step2: Evaluate each option

• Option A: The function cannot have both a global maximum and a global minimum. If \( a>0 \), it has a minimum (not a maximum), if \( a < 0 \), it has a maximum (not a minimum). So A is false.
• Option B: From the analysis above, when \( a>0 \), the function has a global minimum (no global maximum), when \( a < 0 \), the function has a global maximum (no global minimum). So the function has either a global maximum or a global minimum, but not both. B is true.
• Option C: The function has either a global maximum or a global minimum, so C is false.
• Option D: The nature of the global extremum (maximum or minimum) depends only on the sign of \( a \), not on the values of \( b,c,d,k \). So D is false.

Answer:

B. \( f \) has either a global maximum or a global minimum, but not both.