What are the critical values, if any, of #f(x)= 5x + 6x ln x^2#?

Answer 1

# x = e^(-17/12) ~~ 0.659241 \ \ (6 \ dp)#

We have:

# f(x) = 5x+6xln(x^2) #

Which, using the properties of logarithms, we can write as:

# f(x) = 5x+6x(2)ln(x) # # \ \ \ \ \ \ \ = 5x+12xln(x) #
Then, differentiating wrt #x# by applying the product rule we get:
# f'(x) = 5 + (12x)(1/x) + (12)(lnx) # # \ \ \ \ \ \ \ \ \ = 5+12+12lnx # # \ \ \ \ \ \ \ \ \ = 17+12lnx #

At a critical point, we requite that the first derivative vanishes, thus we require that:

# f'(x) = 0 #
# :. 17+12lnx = 0 # # :. lnx = -17/12 # # :. x = e^(-17/12) ~~ 0.659241 \ \ (6 \ dp)#
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Answer 2

To find the critical values of ( f(x) = 5x + 6x \ln(x^2) ), we need to take the derivative of ( f(x) ), set it equal to zero, and solve for ( x ).

The derivative of ( f(x) ) is ( f'(x) = 5 + 6(1 + \ln(x^2)) \cdot \frac{d}{dx}(x^2) ).

Simplify the derivative: ( f'(x) = 5 + 6(1 + \ln(x^2)) \cdot 2x ).

Now, set ( f'(x) ) equal to zero and solve for ( x ): ( 5 + 12x(1 + \ln(x^2)) = 0 ).

( 5 + 12x + 12x \ln(x^2) = 0 ).

Using properties of logarithms, ( \ln(x^2) = 2\ln(x) ):

( 5 + 12x + 24x \ln(x) = 0 ).

There's no direct algebraic way to solve this equation. Critical values can be determined numerically or graphically.

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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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