Find the critical points of the function and use the First Derivative Test to de

Question

Find the critical points of the function and use the First Derivative Test to determine whether the critical point is a local minimum or maximum (or neither). (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) y = x(x - 1)^2 local minimum x = _____ local maximum x = _____ Determine the intervals on which the function is increasing or decreasing. (Enter your answers using interval notation. Enter EMPTY or 0 for the empty set.) increasing _____ decreasing _______

Explanation / Answer

Solution:

y = x(x - 1)^7

Critical numbers are any points on the function where the derivative of the function is equal to zero, or does not exist.

f '(x) = 0:
f'(x) = x d/dx (x-1)^7 + (x-1)^7 d/dx(x)

      = x * 7(x - 1)^6 + (x - 1)^7

     = (x - 1)^6 {7x + x - 1}

     = (x - 1)^6 (8x - 1) = 0

x = 1 and x = 1/8;

critical points are ;

at x = 1 ; y = 0

(x, y) = (1, 0)

first derivative test;

Take any two point let x= -1 and x = 2

y = x(x - 1)^7 = (-1)(-1 - 1)^7 = 128

y = x(x - 1)^7 = (2)(2 - 1)^7 = 2

Positive answers mean the function is increasing, and Negative answers mean the function is decreasing.

Both of the answers found for this function are positive numbers,

which means the graph is rising from negative infinity to 1,

at x = 1 the graph levels off (the derivative equals zero),

then the graph continues to increase from 1 to infinity

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