Please help me with this question.Thank you! Analyze and sketch the graph of the function. Identify any relative extrema, points of inflection, and asymptotes. (Round your answers to three decimal places. If an answerdoes not exist, enter DNE.)In(3x)y =x2relative minimum(х, у) %3relative maximum(х, у) %3inflection point(х, у) %3Dvertical asymptoteX =horizontal asymptoteGraph the function.

Name: Please help me with this question.Thank you! Analyze and sketch the gr
Uploaded: 2024-03-20T07:06:27.000Z
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Description: Please help me with this question.Thank you! Analyze and sketch the graph of the function. Identify any relative extrema, points of inflection, and asymptotes. (Round your answers to three decimal places. If an answerdoes not exist, enter DNE.)In(3x)

Given that, the function is y=ln3xx2. Find the first derivative of y=ln3xx2 as follows, y'x=ddxln3xx2=x2ddxln3xx2-ln3xx2ddxx2x22=1-2ln3xx3 Equate the first derivative to zero to find the critical point. 1-2ln3xx3=01-2ln3x=02ln3x=1x=e3 Therefore, the critical point is x=e3. Substitute x=e3 in y=ln3xx2. ye3=ln3e3e32=92e Therefore, the critical point is e3,92e. Therefore, the intervals of the given function are -∞,e3,e3,∞. Find the second derivative of y=ln3xx2 as follows, y''x=ddx1-2ln3xx3=x3ddx1-2ln3x-1-2ln3xddxx3x32=--6ln3x+5x4 Substitute x=e3 in y''x=--6ln3e3+5e34. y''x=--6ln3e3+5e34=-162e2<0 Maximum Equate the second derivative to zero to find the inflection point. --6ln3x+5x4=06ln3x-5=06ln3x=5x=e563 Therefore, the critical point is x=e563. Substitute x=e563 in y=ln3xx2. y=ln3e563e5632=152e53 Therefore, the inflection point is e563,152e53.In order to find the vertical asymptote, set denominator is equal to 0. x2=0x=0 Therefore, the vertical asymptote is x=0. Note that, If the degree (the largest exponent) of the denominator is bigger than the degree of the numerator, the horizontal asymptote is the x-axis (y = 0). Therefore, the horizontal asymptote is y=0. Use the above values to sketch the graph of the function as shown below,Answers: Relative maximum: e3,92e Relative minimum: = No relative minimum Inflection point: e563,152e53 Vertical asymptote: x=0 Horizontal asymptote: y=0 Graph the function: Sketched.