Maximum and Minimum values

Question

Find the maximum or minimum values of the following quadratic functions, and the values of x for which they occur;

(i) f(x)=x^2-4x+7

(ii) f(x)=3-8x-x^2

Answer 1

take the derivative, set it equal to zero, solve for x

then plug that value of x back into f(x) to solve

for the max or min value of the function

a) f(x) = x^2 -4x+7

f'(x) =2x-4=0

x =2

f(2) = 4-8+7 = 3= minimum

the vertex is the minimum point of an upward opening parabola

(2,3)

b) f(x)= 3-8x-x^2

f'(x) = -2x -8=0

x=8/-2

x=-4

f(-4) = 3+32-16 =19= maximum

the vertex of a downward opening parabola is the maximum point

(-4,19)