Question

5. Please verify correct

Answer 1

Answer: The function has a maximum value

The function's maximum value is 1

The function's maximum value occurs at x=-3

Explanation:

Given equation:
`g(x) = -3x^ - 18x - 26`

We shall use the differentiation method to find the answer.

Differentiating the equation, we get
`g'(x) = - 6x - 18`

equating g'(x) = 0,

`-6x - 18 = 0\\-6x = 18 \\x = -3`

Hence the minimum/ maximum value of the function lies in x = -3

on equating x>-3, g'(x) is negative

on equating x<-3, g'(x) is positive

this proves that first, the function was increasing and after x = -3, the function started to decrease hence at x=-3, the function g(x) was at its maximum value

The function's maximum value at x=-3 is

`g(x) = -3x^2 - 18x - 26\\g(-3) = -3(-3)^2 - 18(-3) - 26\\g(-3) = -3(9) +54 -26\\g(-3) = 54 - 27 -26\\g(-3) = 1`

Answer 2

Our function is a parabola, therefore, it has only one critical point which is its vertex. Since the coefficient of the squared term is negative, the vertex is the maximum point.The critical points of a function are given by the zeros of the first derivative. The first derivative of our function is:

`f^(\prime)(x)=-6x-18`

The solutions for the following equation are the critical points of the original function:

`-6x-18=0\implies x=-3`

The maximum point happens at x = - 3.

To find the corresponding value, we just have to evaluate x = - 3 into our function:

`f(-3)=-3(-3)^2-18(-3)-26=1`

The maximum value of the function is 1, and it occurs at x = - 3.