Consider the following fx x2 14x Find the critical numbers

Consider the following.

f(x) = x² + 14x

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

x =

Find the open intervals on which the function is increasing or decreasing. Use a graphing utility to verify your results. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)

Solution

given f(x) = -x^2 +14x

to find critical points first we have to find f \'(x)

the first derivative of f(x) = -2x +14

critical point is at f\'(x) =0

so -2x +14 =0

2x =14

x=7

so critical number x = 7

finding f\'\'(x)

f\'\'(x) = -2

so this function is increasing for all negative values of x

and it is decreasing for all positive values of x