Consider the following function. g(x) = x2 − 2x − 4 on [0, 9]. The questions are in the description

Question

Find the derivative of the function.

g '(x) =  

 

 

Find any critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

x =  

 

 

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

maximum minimum 

Answer 1

g'(x) = 2x -2 when g(x) =x^2-2x -4

the function is an upward opening parabola

add and subtract 1 to complete the square

y = x^2 -2x + 1 -4 -1 = (x-1)^2 -5

the vertex of the parabola = the absolute & local minimum point = -5 or the point (1,-5)

there is no absolute or local maximum. No real number. there is a "surreal" maximum of infinity but no one means that in this type of problem. they just want to know the real numbers. Infinity is not a real number, it's "just" a limit that the real numbers approach.

the parabola has one turning point which is the vertex or the minimum point (1,-5) It's the critical point where the slope =0 g'(1) = 2(1)-2 = 0. It's where the 2nd derivative g''(x) = 2 >0 which means the curve is concave upward (U shaped) You're at the bottom of the "U" curve at (1,-5) (1,-5) is the point where the slope changes from negative to positive. At x<1, slope is negative, at x>1 slope is positive, at x=0 slope =0