Find the relative minimum of y = 2x 3 + 14x 2 - 10x - 46

The minimum value is 47.704  when x = 1/3 correct to the nearest thousandth.

Step-by-step explanation:

y = 2x^3 + 14x^2 - 10x - 46

To find the turning point(s) on the curve we first differentiate:

dy/dx = 6x^2 + 28x - 10   This = zero  for turning points:

6x^2 + 28x - 10 = 0

2(3x^2 + 14x - 5) = 0

(3x - 1 )(x + 5) = 0

x= 1/3, - 5.

We now determine the nature of the turning points:

The second derivative  is 12x + 28

When x = 1/3  it's value is positive.

When x = -5  its value is -60 + 28 which is  negative.

So x = 1/3 gives a relative minimum.

When x = 1/3, y = -47.704.

question: how many people can she serve?

answer: she can serve 16.66666667 people. so she can only give 16 people a complete serving.

question: does she have enough ice cream for her guests?

answer: no. explanation: she only has enough for 16 people to have a full serving. however, she can give a 17th person a partial serving.