Find the x-value of all points where the function below has any relative extrema

Question

Find the x-value of all points where the function below has any relative extrema. Find the value(s) of any relative extrema. G(x) = x^10 e^X - 4 Select the correct choice below and. if necessary, fill in the answer box to complete your choice. The function has a relative minimum at the point(s) (0. -4). (Simplify your answer. Type an ordered pair Use integers or decimals for any numbers in the expression. Round to two decimal places as needed. Use a comma to separate answers as needed.) The function has no relative minimum. Select the correct choice below and. if necessary, fill in the answer box to complete your choice. The function has a relative maximum at the point(s) (Simplify your answer Type an ordered pair. Use integers or decimals for any numbers in the expression Round to two decimal places as needed. Use a comma to separate answers as needed.) The function has no relative maximum.

Explanation / Answer

G(x) = x^10 * e^x - 4

dG/dx = x^10 * e^x + e^x * 10*x^9

make dG/dx = 0 to find inflextion points

x^9* e^x ( x+10) = 0

so solving , we have inflection points as x= 0 and x = -10

to find at which point maxima and minima occurs ,we need to find d^2G/dx^2

d^2G/dx^2 = x^10 * e^x + 10x^9 * e^x + 10( x^9 e^x +e^x * 9 *x^8)

d^2G/dx^2 = x^8 *e^x ( x^2 +20x+90)

at x =0 , d^2G/dx^2 = 0

and at x = -10 , d^2G/dx^2 = (-10)^8 *e^(-10) ( (-10)^2 +20(-10)+90) = -45399.9297625

this is <0 so at x= -10 , we have maximum

as you have already figured out minimum , i'll just calclate maximum only

at x= -10 , G(x) = (-10)^10 * e^(-10)-4 = 453995.30

so maximum point is (-10,453995.30) ------ANSWER

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