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If f(x) is a real and continuous function of x, the Taylor series expansion of f(x) about
its minima will NEVER have a term containing
its minima will NEVER have a term containing
- a)first derivative
- b)second derivative
- c)third derivative
- d)any higher derivative
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
If f(x) is a real and continuous function of x, the Taylor series expa...
So, Taylor series expansion of f(x) about ‘a’ will never contain first derivative term.
Most Upvoted Answer
If f(x) is a real and continuous function of x, the Taylor series expa...
Taylor Series Expansion of a Function
The Taylor series expansion is a way to approximate a function using a polynomial. It is based on the idea that any function can be expressed as an infinite sum of terms, each term representing a derivative of the function evaluated at a specific point. The Taylor series expansion of a function f(x) about a point x=a is given by:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
Here, f'(a) represents the first derivative of f(x) evaluated at x=a, f''(a) represents the second derivative, f'''(a) represents the third derivative, and so on.
Expanding about the Minima
When we expand the Taylor series about the minima of a function, we are essentially expanding it about a point where the first derivative is zero. This is because the minima of a function occur where the slope (first derivative) changes from positive to negative.
Analysis of Options
a) First Derivative: The Taylor series expansion about the minima will have a term involving the first derivative of the function. This is because the first derivative is zero at the minima, and it contributes to the expansion.
b) Second Derivative: The Taylor series expansion about the minima will have a term involving the second derivative of the function. This is because the second derivative provides information about the curvature of the function, which is relevant at the minima.
c) Third Derivative: The Taylor series expansion about the minima will have a term involving the third derivative of the function. This is because the third derivative provides information about the rate of change of curvature of the function, which is relevant at the minima.
d) Higher Derivatives: The Taylor series expansion about the minima will have terms involving higher derivatives of the function. These higher derivatives provide information about the higher-order behavior of the function, which can affect the shape of the function near the minima.
Conclusion
From the analysis above, we can see that the Taylor series expansion of f(x) about its minima will include terms involving the first derivative, second derivative, third derivative, and higher derivatives. Therefore, the correct answer is option 'A' - the expansion will always have a term involving the first derivative.
The Taylor series expansion is a way to approximate a function using a polynomial. It is based on the idea that any function can be expressed as an infinite sum of terms, each term representing a derivative of the function evaluated at a specific point. The Taylor series expansion of a function f(x) about a point x=a is given by:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
Here, f'(a) represents the first derivative of f(x) evaluated at x=a, f''(a) represents the second derivative, f'''(a) represents the third derivative, and so on.
Expanding about the Minima
When we expand the Taylor series about the minima of a function, we are essentially expanding it about a point where the first derivative is zero. This is because the minima of a function occur where the slope (first derivative) changes from positive to negative.
Analysis of Options
a) First Derivative: The Taylor series expansion about the minima will have a term involving the first derivative of the function. This is because the first derivative is zero at the minima, and it contributes to the expansion.
b) Second Derivative: The Taylor series expansion about the minima will have a term involving the second derivative of the function. This is because the second derivative provides information about the curvature of the function, which is relevant at the minima.
c) Third Derivative: The Taylor series expansion about the minima will have a term involving the third derivative of the function. This is because the third derivative provides information about the rate of change of curvature of the function, which is relevant at the minima.
d) Higher Derivatives: The Taylor series expansion about the minima will have terms involving higher derivatives of the function. These higher derivatives provide information about the higher-order behavior of the function, which can affect the shape of the function near the minima.
Conclusion
From the analysis above, we can see that the Taylor series expansion of f(x) about its minima will include terms involving the first derivative, second derivative, third derivative, and higher derivatives. Therefore, the correct answer is option 'A' - the expansion will always have a term involving the first derivative.
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If f(x) is a real and continuous function of x, the Taylor series expansion of f(x) aboutits minima will NEVER have a term containinga)first derivativeb)second derivativec)third derivatived)any higher derivativeCorrect answer is option 'A'. Can you explain this answer?
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