Calculate Functions

Question

Given the following functions f(x)=x^2+5x+7 evaluate:

a. f'(-3)

b. f'' (9)

Answer 1

f(x) =x^2 +5x + 7

f'(x) = 2x +5

f'(-3) =2(-3)+5 = -6+5= -1

f"(x) = 2

f"(9) = 2

the derivative of any real number = 2

the function is an upward opening parabola with y intercept = 7, there are no x intercepts

vertex =(-b/2a, (-b/2a)^2 +5(-b/2a) +7) = (-5/2, 3/4)

at x=-5/2, f'(x) = 0

for x<-5/2, f'(x)<0

the parabola is concave upwards everywhere, so f"(x) is always positive

the 2nd derivative is the rate of change of the 1st derivative