بحث دالة بولينوم – سؤال 3

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بحث دالة كثيرة الحدود (بولينوم)
ابحث الدالة:
𝒚 = 𝟎. 𝟓𝒙𝟒 – 𝟖𝒙𝟐
.1مجال التعريف:
الدالة معرفة لكل 𝑥.
.2نقاط التقاطع مع المحاور:
تقاطع الدالة مع محور 𝑥 نعوض : 𝑦 = 0
0 = 0.5𝑥4 – 8𝑥2 = 𝑥2 ∙ (0.5𝑥2 – 8) = 0 → 𝑥 = 0 , 𝑥 = -4 , 𝑥 = 4
لذلك نقاط التقاطع مع محور 𝑥 هي ).(-4,0) , (4,0) , (0,0
تقاطع الدالة مع محور 𝑦 نعوض : 𝑥 = 0
𝑦 = 0.5 ∙ 04 – 8 ∙ 02 = 0 → 𝑦 = 0
لذلك نقاط التقاطع مع محور 𝑦 هي: ).(0,0
.3نقاط قصوى وتحديد نوعها (:)Max/Min
نشتق الدالة: 𝑥𝑦= 2𝑥3 – 16
نساوي المشتقة لصفر ثم نحل المعادلة التي تنتج:
𝑦= 0 → 2𝑥3 – 16𝑥 = 0 → 𝑥 ∙ (2𝑥2 – 16) = 0 → 𝑥 = 0 , 𝑥 = -2.828 , 𝑥 = 2.828
نعوض 𝑥 = 0 , 𝑥 = -2.828 , 𝑥 = 2.828في الدالة لكي نجد الاحداثي 𝑦:
𝑦 = 0.5 ∙ 04 – 8 ∙ 02 = 0 → (0,0) نقطة شك
𝑦 = 0.5 ∙ 2.8284 – 8 ∙ 2.8282 = -32 → (2.828, -32) نقطة شك
𝑦 = 0.5 ∙ (-2.828)4 – 8 ∙ (-2.828)2 = -32 → (-2.828, -32) نقطة شك
نبني الجدول لكي نحدد نوع النقطة (وايضا لايجاد مجالات التصاعد والتنازل:)

نقطة
اختبار
𝑥 = 3
نقطة شك
𝑥 = 2.828
نقطة اختبار
𝑥 = 1
نقطة شك
𝑥 = 0
نقطة
اختبار
𝑥 = -1
نقطة شك
𝑥 = -2.828
نقطة اختبار
𝑥 = -3
نعوض نقاط الاختبار في
المشتقة
+ + اشارة المشتقة )𝑋(𝑓
تصرف الدالة )𝑋(𝑓
Min
(2.828, -32)
Max
(0,0)
Min
(-2.828, -32)
استنتاج عن الدالة )𝑋(𝑓


يمكن ايضا ان نفحص نوع النقاط القصوى عن طريق المشتقة الثانية: 𝑦′′ = 6𝑥2 – 16
𝑦′′(0) = 6 ∙ 02 – 16 = -16 < 0 → Max (0,0)
𝑦′′(2.828) = 6 ∙ 2.8282 – 16 = 32 > 0 → Min (2.828, -32)
𝑦′′(-2.828) = 6 ∙ (-2.828)2 – 16 = 32 > 0 → Min (-2.828, -32)
نقاط قصوى مطلقة: يمكن ان نستنتج ان للدالة يوجد نقاط صغرى مطلقة وهي:
) Min (-2.828, -32) , Min (2.828, -32لان الدالة تحصل على اصغر قيمة
ممكنة لها في هذة النقاط . لا يوجد نقاط عظمى مطلقة للدالة لان قيمها من الاعلى غير
محدودة (قيم
𝑦 غير محدودة).
.4مجالات تصاعد ومجالات تنازل الدالة:
نستنتج المجالات من الجدول اعلاه:
مجال التنازل: .
𝑥 < -2.828 , 0 < 𝑥 < 2.828
مجال التصاعد: . -2.828 < 𝑥 < 0 , 𝑥 > 2.828
.5مجالات موجبة ومجالات سالبة:
نعوض في الدالة 𝑦 = 0لكي نجد النقاط الصفرية للدالة: لقد وجدنا النقاط الصفرية
في بند
2ووجدنا انها: , 𝑥 = 0 , 𝑥 = 4 , 𝑥 = -4نضع هذة النقاط على محور 𝑥 ثم
نختار نقاط اختبار مناسبة, نعوضها في الدالة ونحدد اشارة الدالة:
لذلك نستنتج ان: المجال الموجب هو: .
𝑥 < -4 , 𝑥 > 4
المجال السالب هو: .-4 < 𝑥 < 0 , 0 < 𝑥 < 4

-4 0 4

X
-5 -1 1
اشارة الدالة (موجب/سالب)
نقاط اختبار (نعوضها في الدالة)
5
.6رسم بياني تقريبي:
حسب البنود السابقة الرسم البياني التقريبي للدالة هو:
X
Y
Min
Max
Min


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