Let f (x )equals x squared minus 10 x plus 6. a. Find the values of x for which the slope of the curve yequalsf(x) is 0. b. Find the values of x for which the slope of the curve yequalsf(x) is negative 6.

a) x = 5

b) x = 2

Step-by-step explanation:

The given function is  f(x) = x² - 10x + 6 and we want to find the values of x such that slope of the curve equals to 0 and -6. This can be done by taking the derivative of the function f(x) and setting that equal to 0 and -6 respectively.

When slope of curve = 0

Take the derivative of the function f(x)

f(x) = x² - 10x + 6

f'(x) = 2x - 10 + 0

f'(x) = 2x - 10

Now put f'(x) = 0

0 = 2x - 10

2x = 10

x = 10/2 = 5

Hence x = 5

When slope of curve = -6

We have already taken the derivative so we just have to put f'(x) = -6

-6 = 2x - 10

2x = -6 + 10

2x = 4

x = 4/2 = 2

Hence x = 2

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1