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MOTORGEEK MOTORGEEK
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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?
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Calculus: Early Transcendentals


Edition: 3rd
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jaray75jaray75
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More questions for this book are available here
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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