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If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =?
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If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to...
Problem:
If \(f(x) = ax^2 + bx + c\) and \(a, b, c\) belong to real numbers and \(a\) is not equal to zero. Let \(f(x) > 0\) for all \(x\). Find \(g(x)\) where \(g(x) = f(x)f'(x)f''(x)\).
Solution:
We are given that \(f(x) = ax^2 + bx + c\) and \(f(x) > 0\) for all \(x\). We need to find \(g(x) = f(x)f'(x)f''(x)\).
Step 1: Find the derivative of \(f(x)\)
To find the derivative of \(f(x)\), we differentiate each term separately using the power rule of differentiation:
\(f'(x) = \frac{d}{dx}(ax^2) + \frac{d}{dx}(bx) + \frac{d}{dx}(c)\)
\(f'(x) = 2ax + b + 0\)
Simplifying, we get:
\(f'(x) = 2ax + b\)
Step 2: Find the second derivative of \(f(x)\)
To find the second derivative of \(f(x)\), we differentiate \(f'(x)\) with respect to \(x\):
\(f''(x) = \frac{d}{dx}(2ax + b)\)
\(f''(x) = 2a\)
Step 3: Substitute the values into \(g(x)\)
Now that we have the expressions for \(f(x)\), \(f'(x)\), and \(f''(x)\), we can substitute them into \(g(x) = f(x)f'(x)f''(x)\):
\(g(x) = (ax^2 + bx + c)(2ax + b)(2a)\)
Simplifying further, we get:
\(g(x) = 4a^3x^3 + 2a^2bx^2 + 2abx^2 + ab^2x + 2acx + cb\)
Conclusion:
The expression for \(g(x)\) is \(4a^3x^3 + 2a^2bx^2 + 2abx^2 + ab^2x + 2acx + cb\). This is the final answer to the problem.
If \(f(x) = ax^2 + bx + c\) and \(a, b, c\) belong to real numbers and \(a\) is not equal to zero. Let \(f(x) > 0\) for all \(x\). Find \(g(x)\) where \(g(x) = f(x)f'(x)f''(x)\).
Solution:
We are given that \(f(x) = ax^2 + bx + c\) and \(f(x) > 0\) for all \(x\). We need to find \(g(x) = f(x)f'(x)f''(x)\).
Step 1: Find the derivative of \(f(x)\)
To find the derivative of \(f(x)\), we differentiate each term separately using the power rule of differentiation:
\(f'(x) = \frac{d}{dx}(ax^2) + \frac{d}{dx}(bx) + \frac{d}{dx}(c)\)
\(f'(x) = 2ax + b + 0\)
Simplifying, we get:
\(f'(x) = 2ax + b\)
Step 2: Find the second derivative of \(f(x)\)
To find the second derivative of \(f(x)\), we differentiate \(f'(x)\) with respect to \(x\):
\(f''(x) = \frac{d}{dx}(2ax + b)\)
\(f''(x) = 2a\)
Step 3: Substitute the values into \(g(x)\)
Now that we have the expressions for \(f(x)\), \(f'(x)\), and \(f''(x)\), we can substitute them into \(g(x) = f(x)f'(x)f''(x)\):
\(g(x) = (ax^2 + bx + c)(2ax + b)(2a)\)
Simplifying further, we get:
\(g(x) = 4a^3x^3 + 2a^2bx^2 + 2abx^2 + ab^2x + 2acx + cb\)
Conclusion:
The expression for \(g(x)\) is \(4a^3x^3 + 2a^2bx^2 + 2abx^2 + ab^2x + 2acx + cb\). This is the final answer to the problem.
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If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =?
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If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =?.
If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If f (x)=ax^2 bx c and a,b,c belong to real numbers and a not equal to zero Let f (x)>0 for all x G (x)= f (x) f' (x) f" (x) Then g (x ) =?.
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