can i have the answer please Find an equation for the line tangent to the graph

Question

can  i have the answer please

Find an equation for the line tangent to the graph of f(x) = x2 ln x at x = e3. Suppose f(x) = ln(x2 + l). Calculate the first and second derivatives of f. Determine the intervals where f is increasing or decreasing. Determine all local maxima and minima for f. Determine the intervals where f is concave up or concave down. Determine all points of inflection for f. Using (a)-(e), and plotting two or three points on the graph, sketch a graph of f. Evaluate the indefinite integral Evaluate the definite integral You will probably want to use the method of substitution (aka change of variables) to do this problem. If you do, be sure to remember that when substitution is used for a definite integral, the limits of integration have to be changed to the correct ones for the new variable. (Bonus question): Determine the derivative of f(x) = (ln x)ln x. (Logarithmic differentiation will simplify the compulations considerably.)

Explanation / Answer

1) f(x) = x^2 lnx

f'(x) = 2x lnx + x^2 *1/x = 2x lnx + x

at x= e^3

slope = f'(x) = 2*e^3 ln(e^3) + e^3 = 6e^3 *1 + e^3 = 7 e^3

at x= e^3 , y = e^6 ln(e^3) = 3e^6

(e^3, 3e^6) , m = 7e^3

tangent equation is

y - y1 = m(x-x1)

y - 3e^6 = 7e^3 ( x- e^3 )

y = 7e^3 x - 4 e^6

2) f(x) = ln(x^2 +1)

f'(x) = 1/(x^2 +1) * 2x = 2x/(x^2 +1)

f"(x) ={ 2*(x^2 +1) - 2x(2x) }/(x^2 +1)^2

= ( 2x^2 + 2 - 4x^2 )/(x^2 +1)^2

= ( 2-2x^2)/(1+x^2)^2

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