How do you find the second derivative test to find extrema for #f(x) = 2x^2lnx-5x^2#?

Answer 1

See the explanation.

#D_f=R^+#
#f'=4xlnx+2x^2*1/x-10x=4xlnx-8x=4x(lnx-2)#
#f''=4lnx+4x*1/x-8=4lnx-4=4(lnx-1)#
#f'=0 <=> 4x=0 vv lnx-2=0#
#x=0 !in D_f# #lnx=2 => x=e^2#
#f''(e^2)=4(lne^2-1)=4(2-1)=4>0# and hence function has a minimum at #x=e^2#.
#f_min=f(e^2)=2(e^2)^2lne^2-5(e^2)^2=4e^4-5e^4=-e^4#
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Answer 2

To find extrema for ( f(x) = 2x^2 \ln(x) - 5x^2 ), follow these steps:

  1. Find the first derivative of ( f(x) ) using the product rule: ( f'(x) = 4x \ln(x) + 2x - 5 ).
  2. Find critical points by setting ( f'(x) ) equal to zero and solving for ( x ).
  3. Once you have critical points, use the second derivative test to determine whether each critical point corresponds to a relative maximum, relative minimum, or neither.
  4. Find the second derivative of ( f(x) ): ( f''(x) = 4\ln(x) + 6/x ).
  5. Evaluate ( f''(x) ) at each critical point found in step 2.
  6. If ( f''(x) > 0 ) at a critical point, then it corresponds to a relative minimum.
  7. If ( f''(x) < 0 ) at a critical point, then it corresponds to a relative maximum.
  8. If ( f''(x) = 0 ) or is undefined at a critical point, the test is inconclusive.

That's the process for using the second derivative test to find extrema for the given function.

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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