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.If a spring stretches 0.9 m when a 6-kg weight is attached to it, how much will it stretch when a 14-kg weight is attached to it? (Round to the nearest tenth.)
A. 5.1 m B. 1.1 m C. 2.1 m D. 4.1 m
Evaluate the integral.
A. - 
B. 
- 
C. 
- 
D. 
- 
Factor the sum or difference of two cubes completely.729 - x3
A. (9 + x)(81 - x2) B. (9 - x)(81 + x2) C. (9 - x)(81 + 9x + x2) D. (9 + x)(81 - 9x + x2)
Find the maximum or minimum point of the function and state whether it is a maximum or minimum.f(x) = x2 + 12x + 33
A. (-6, -3); minimum B. (-3, -6); maximum C. (-3, 0); maximum D. (0, -6); minimum