Consider the following function.

Question

g(x) = x² − 2x − 6 on [0, 7]

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

We can find g'(x) easily by using the rule that d/dx(xⁿ) = n*x^n-1

g'(x) = 2x - 2 **the derivative of a constant is zero so the 6 goes away when we differentiate

To find the critical numbers we set the derivative equal to zero

g'(x) = 0 = 2x-2 *solving for x we get x=1 as our critical number.

When we check for absolute minima/maxima we can't forget to also check the end points of our interval which are x=7 and x=0 we find which is which by plugging our endpoints and critcal number back into the original function g(x)

g(0) = -6

g(1) = -7

g(7) = 29

absolute maximum on this interval = 29

absolute minimum on this interval = -7