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On which domain restriction will (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^2+2x+2), become decreasing and one-one?फलन (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^2+2x+2) को किस प्रांत पर सीमित करने से यह घटता हुआ और एकैकी बनेगा?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \((-\infty,-1]\) / ((-\infty,-1])\((-\infty,-1]\) / ((-\infty,-1])
Step 1: (f(x)=(x+1)^2+1), so the vertex is at (x=-1). Step 2: On ((-\infty,-1]), the function is strictly decreasing and hence one-one. Step 3: Remember that a quadratic decreases to the left of its vertex and increases to the right.चरण 1: (f(x)=(x+1)^2+1), इसलिए शीर्ष (x=-1) है। चरण 2: ((-\infty,-1]) पर यह फलन लगातार घटता है और इसलिए एकैकी है। चरण 3: द्विघात फलन में शीर्ष के बाईं ओर घटता और दाईं ओर बढ़ता व्यवहार याद रखें।
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