Question

(a) The line y=b+mx is tangent to the graph of f at x=3. Given that f(3)−9, the derivative f(3)=4, and f(3)=−2, find m and b. m= b=

(b) Near x=3, does the graph of flie above or below the line from part (a)? The graph of flies the line

Answer 1

The slope m of the tangent line is 4, and the y-intercept b is -3, making the equation of the line y = -3 + 4x. The graph of f lies below the tangent line just after x=3 because the second derivative f''(3) is negative.

Step-by-step explanation:

To find the slope m and y-intercept b of a tangent line to the function f at x=3, we use the given information: f(3) = 9, f'(3) = 4, and the general equation of the line y = b + mx. Since the line is tangent to the graph of f at x=3, and the derivative at that point is 4, this means the slope m of the tangent line is also 4. Using f(3) = 9, we can plug in x = 3 and y = 9 into the line equation to find b:

y = b + mx
9 = b + 4(3)
9 = b + 12
b = 9 - 12
b = -3

Therefore, the equation of the tangent line is y = -3 + 4x. To determine whether the graph of f lies above or below this line near x=3, we examine the sign of the second derivative f''(3) = -2. A negative second derivative indicates the graph of f is concave down, so the graph lies below the tangent line just after x=3.