What are the points of inflection of #f(x)=x^{2}e^{11 x} #?
Sufficient condition for giving points of inflexion, if any#:r
f'' is not zero and f''=0.
Here,
f''=0 results in
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To find the points of inflection for the function ( f(x) = x^2 e^{11x} ), we need to locate where the concavity changes. Points of inflection occur where the second derivative changes sign.

Find the first derivative of ( f(x) ): [ f'(x) = 2x e^{11x} + x^2 (11e^{11x}) ]

Find the second derivative of ( f(x) ): [ f''(x) = 2e^{11x} + 2x(11e^{11x}) + 22x e^{11x} ]

Set ( f''(x) ) equal to zero and solve for ( x ): [ 2e^{11x} + 2x(11e^{11x}) + 22x e^{11x} = 0 ] [ e^{11x}(2 + 22x + 22x^2) = 0 ] Since ( e^{11x} ) is always positive, we solve for the polynomial inside the parentheses: [ 2 + 22x + 22x^2 = 0 ]

Solve the quadratic equation ( 22x^2 + 22x + 2 = 0 ) to find the values of ( x ).
After finding the values of ( x ), evaluate the concavity of ( f(x) ) around these points to determine whether they are points of inflection.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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