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

The question is:

Let f (x) = x² - 10x + 6

a. Find the values of x for which the slope of the curve y = f (x) is 0.

b. Find the values of x for which the slope of the curve y = f (x) is - 6.

Answer:

(a) The value of x for which the slope is 0 is 5

(b) The value of x for which the slope is - 6 is 2

Step-by-step explanation:

Given y = f (x) = x² - 10x + 6

The Slope is given as dy/dx = f' (x)

It is obtained by differentiating y = f (x) once with respect to x.

Let us find the first derivative of

y = x² - 10x + 6

That is

dy/dx = 2x - 10

Now

(a) We want to find the value of x for which the slope dy/dx = 0

We set

2x - 10 = 0

And solve for x, to have

2x = 10

x = 10/2 = 5

(b) The value of x for which the slope is - 6

We set

2x - 10 = - 6

2x = - 6 + 10 = 4

x = 4/2 = 2