Provide an appropriate response.Consider the quartic function f(x) = ax4 + bx3 + cx2 + dx + e, a ? 0. Must this function have at least one critical point? Give reasons for your answer. (Hint: Must 
for some x?) How many local extreme values can f have?
What will be an ideal response?
Yes. The derivative is cubic: 4ax3 + 3bx2 + 2cx + d. The derivative approaches -? as x approaches 
and it approaches ? as x approaches ?. By the Intermediate Value Theorem, f'(x) must equal zero at at least one point on the interval -? ? x ? ?.
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Solve the problem.Find the tangent line(s) at the pole for the curve 
A. ? = 0 (y = 0), ? = 
(y = x), ? = 
(x = 0), ? = 
(y = -x)
B. ? = 
[y = 
x], ? = 
[y = - 
x]
C. ? = 0 (y = 0), ? = 
(x = 0)
D. ? = 
(y = x), ? = 
(y = -x)
Solve the indicated system of equations algebraically.The product of the lengths of the legs of a right triangle is 108. The hypotenuse has a length of 6
. Find the lengths of the legs.
A. 6, 6 B. 18, 18 C. 6, 18 D. 19, 6
Find the minimum-cost spanning tree for the graph.
A. 
B. 
C. 
D. 
Solve the problem.In recent years, many states have passed laws against smoking in public buildings. The total number of states N that have passed a no smoking in public buildings law, t years after 1990 is given by the function, N(t) = 
. How many states had passed the law in 1990?
A. 0 B. 0.313 C. 1.563 D. 3.125