# Calculus---derivative

Explain why the cubic polynomial function

f(x)=ax^(3) + bx^(2) + cx +d (a unequal 0)

can have either two, one, or no critical points on the real line.

Produce examples that illustrate each of the three cases.

### 3 個解答

• Wen
Lv 5
7 年前
最愛解答

觀察critical的特性，會與f'(x)的 root有關

f'是一個二次函數，所以它的root會有相異根、重根、無實根。

2013-10-16 12:31:35 補充：

f'(x) = 3ax^2 + 2bx + c

if f'(t) = 0 or not exist, then we call that t is critical point.

Since f'(x) is polynomial, so f'(x) is exist on R

=> no one t such that f'(t) is not exist.

consider f'(x) = 0

i.e ax^2 + 2bx + c = 0

x = (-2b ± √D)/2a, where D = (4b^2 - 4ac)

if f have two critical pts, the mean is that roots of f'(x) are distinct.

=>D > 0

if f has one critical pt, the mean is that roots of f'(x) are same

=>D = 0

if f has no critical pt, the mean is that f'(x) hsa no real root

=>D < 0

Examples:two: Let b=2, a = c = 1 => D = 12 > 0，d is arb number, we can let d = 0

one: Let a = b = a = 1 => D = 0, d is arb number, we can let d = 3

no: let b = 0, a = c = 1 => D = -4 <０, d is arb number, we can let d = -9

Done.

• 匿名
7 年前

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• 7 年前

所以講到尾要同佢講 fundamental theorem of algebra.