Question
Download Solution PDFउस अंतराल का पता लगाएं जिसमें फलन f(x) = 2x3 - 24x + 5 बढ़ता हुआ है।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
माना कि f(x) अंतराल (a, b) पर परिभाषित एक फलन है, इस फलन को पूर्ण रूप से एक बढ़ता हुआ फलन कहा जाता है:
- यदि x1 < x2 तो f(x1) < f(x2) ∀ x1, x2 ∈ (a, b)
- यहाँ, \(\frac{{dy}}{{dx}} > 0\;or\;f'\left( x \right) > 0\)
उसीप्रकार, f(x) को एक घटता हुआ फलन कहा जाता है:
- यदि x1 < x2 तो f(x1) > f(x2) ∀ x1, x2 ∈ (a, b)
- यहाँ, \(\frac{{dy}}{{dx}} < 0\;or\;f'\left( x \right) < 0\)
गणना:
दिया हुआ: f(x) = 2x3 - 24x + 5
यहां हमें उस अंतराल को खोजना होगा जिसमें f(x) बढ़ता हुआ है।
पहले f(x) की गणना करें
⇒ f'(x) = 6x2 - 24
जैसा कि हम जानते हैं कि बढ़ते हुए फलन f(x) के लिए हमारे पास f'(x) > 0 है
⇒ 6x2 - 24 > 0
⇒ x2 ≥ 4
⇒ x ∈ (- ∞, - 2) ∪ (2, ∞)
इसलिए जिस अंतराल में फलन बढ़ता हुआ है वह (- ∞, - 2) ∪ (2, ∞) है
Last updated on Jun 20, 2025
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