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If (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^3+x^2+x+1), what is the status of one-one nature of (f)?यदि (f:\mathbb{R}\to\mathbb{R}), (f(x)=x^3+x^2+x+1) है, तो (f) के एकैकी होने की स्थिति क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: It is one-oneयह एकैकी है
Step 1: (f'(x)=3x^2+2x+1). Step 2: Its discriminant is (2^2-4\cdot3\cdot1=-8<0) and the leading coefficient is positive, so (f'(x)>0) for all (x). Step 3: If the derivative is always positive, the function is one-one.चरण 1: (f'(x)=3x^2+2x+1) है। चरण 2: इसका विविक्तकर (2^2-4\cdot3\cdot1=-8<0) और अग्र गुणांक धनात्मक है, इसलिए (f'(x)>0) हर (x) पर। चरण 3: अवकलज हमेशा धनात्मक हो तो फलन एकैकी होता है।
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