Final answer:
To find the point on the curve y = 6 + 2x where the tangent line is perpendicular to the line 16x + 4y = 1, we can solve for x and find the corresponding y-value.
Step-by-step explanation:
To find the point on the curve y = 6 + 2x where the tangent line is perpendicular to the line 16x + 4y = 1, we need to find the slope of the tangent line first.
The slope of the line 16x + 4y = 1 can be determined by rearranging the equation to slope-intercept form, y = mx + b. Here, 16x + 4y = 1 becomes y = -4x + 1/4. Therefore, the slope of the line is -4.
Since perpendicular lines have slopes that are negative reciprocals of each other, the slope of the tangent line will be 1/4. By setting 2 as the derivative of y = 6 + 2x equal to 1/4, we can solve for x. This gives us x = -3/8.
Substituting the value of x into the equation of the curve, we can find the corresponding y-value. Thus, at the point (-3/8, 5), the tangent line is perpendicular to the line 16x + 4y = 1.