Answer:
the correct answer is C. 12.
Step-by-step explanation:
To find the value of k, we need to find the point of tangency between the line and the graph.
The equation of the line is given as 7x + y = k, and the equation of the graph is y = 9x + 1/x².
To find the point of tangency, we need to equate the slopes of the line and the graph.
The slope of the line can be determined by rearranging the equation in slope-intercept form (y = mx + b), where m represents the slope. In this case, the equation becomes y = -7x + k.
The slope of the graph can be determined by differentiating the equation with respect to x. Differentiating y = 9x + 1/x² gives us dy/dx = 9 - 2/x³.
Now, equating the slopes, we have:
-7 = 9 - 2/x³
Simplifying the equation, we get:
-7x³ = 9x³ - 2
16x³ = 2
Dividing both sides by 16, we have:
x³ = 2/16
x³ = 1/8
Taking the cube root of both sides, we get:
x = 1/2
Now, substituting the value of x into the equation of the line, we have:
y = -7(1/2) + k
y = -7/2 + k
Since the line is tangent to the graph, the point of tangency will have the same x-coordinate on both the line and the graph.
Substituting x = 1/2 into the equation of the graph, we have:
y = 9(1/2) + 1/(1/2)²
y = 9/2 + 4
y = 17/2
Therefore, the point of tangency is (1/2, 17/2).
Substituting this point into the equation of the line, we have:
17/2 = -7/2 + k
Adding 7/2 to both sides, we get:
17/2 + 7/2 = k
24/2 = k
12 = k
Hence, the value of k is 12.
Therefore, the correct answer is C. 12.
