Question

Find a number a such that the tangent lines to

y=x+7x²+x+12 at x=a and at x=a+1 are parallel.
a)a=0
b)a=−1
c)a=1
d)a=2

Answer 1

To find a number a such that the tangent lines to y = x + 7x² + x + 12 at x = a and at x = a + 1 are parallel, we need to find the slope of the tangent lines at these points. However, there is no number a that satisfies this condition.

Step-by-step explanation:

To find a number a such that the tangent lines to y = x + 7x² + x + 12 at x = a and at x = a + 1 are parallel, we need to find the slope of the tangent lines at these points. The slope of a tangent line to a function is given by the derivative of the function evaluated at the specific x-value. So, we need to find the derivative of y = x + 7x² + x + 12 and set it equal to the same value for x = a and x = a + 1. Let's find the derivative first:

y' = 1 + 14x + 1 = 14x + 2

Now, we set the derivative equal to the same value:

14a + 2 = 14(a + 1) + 2

14a + 2 = 14a + 14 + 2

14a + 2 = 14a + 16

Subtracting 14a from both sides:

2 = 16

This equation is not true for any value of a. Therefore, there is no number a such that the tangent lines to y = x + 7x² + x + 12 at x = a and at x = a + 1 are parallel.