How do you find the critical points for #x^4(lnx) #?

Answer 1

#P_min(e^(-1/4),-1/(4e))#

we get #f'(x)=x^3(4ln(x)+1)# and since we have #x>0# we get #f'(x)=0# if #x=e^(-1/4)# so #f''(x)=3x^2(4ln(x)+1)+x^3+4/x#
#f''(x)=3x^2(4ln(x)+1)+4x^2#

and

#f''(e^(-1/4))=4(e^(-1/4))^2>0# and #f(e^(-1/4))=-1/4*e^(-1)#
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Answer 2

To find the critical points of the function f(x) = x^4(lnx), follow these steps:

  1. Find the first derivative of the function.
  2. Set the first derivative equal to zero and solve for x.
  3. Check for critical points where the derivative is undefined (e.g., at x = 0).
  4. Evaluate the second derivative at the critical points to determine their nature (whether they correspond to maxima, minima, or points of inflection).

Let's proceed with the steps:

  1. Find the first derivative: f'(x) = 4x^3(lnx) + x^2(1/x) = 4x^3(lnx) + x

  2. Set the first derivative equal to zero and solve for x: 4x^3(lnx) + x = 0

  3. Since x ≠ 0 (as ln(0) is undefined), we can divide both sides of the equation by x: 4x^2(lnx) + 1 = 0

  4. Solve for x. This equation may not have an algebraic solution, so you may need to use numerical methods or graphical analysis to find approximate solutions.

  5. Once you have the critical points, you can evaluate the second derivative at each critical point to determine their nature (whether they correspond to maxima, minima, or points of inflection). If the second derivative is positive, the point is a local minimum; if it's negative, it's a local maximum; and if it's zero, the test is inconclusive (you may need to use other methods to determine the nature of the critical point).

That's the process for finding the critical points of the function f(x) = x^4(lnx).

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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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