Find the point(s) on the graph of f where the tangent line is horizontal. (If an answer does not exist, enter DNE.)

Question

f(x) = (x² + 8)(7 − x).

find smaller x-value (x,y)=

find larger x-value (x,y)=

Answer 1

take the derivative and set it equal to zero

f(x) = -x^3 +7x^2 -8x +56

f'(x) =-3x^2 +14x -8 = 0

3x^2 -14x + 8 = 0

(3x-2)(x-4) =0

x = 2/3 or 4

y = (2/3^2+8)(7+2/3) = 76/9 + 69/9 = 145/9

(x,y) = (2/3, 145/9)

or x=4, then y = (72)(3) = 216

(x,y) = (4,216)