Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).
Find an equation of the line tangent to y = h(x) at x = 3.

Verified step by step guidance

1. Step 1: Identify the function h(x) and its derivative. We have h(x) = 3f(x). To find the derivative h'(x), use the constant multiple rule: h'(x) = 3f'(x).
2. Step 2: Evaluate h(x) and h'(x) at x = 3. We know f(3) = 1 and f'(3) = 4. Therefore, h(3) = 3f(3) = 3 * 1 = 3 and h'(3) = 3f'(3) = 3 * 4 = 12.
3. Step 3: Use the point-slope form of the equation of a line to find the tangent line. The point-slope form is y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) is a point on the line.
4. Step 4: Substitute the values into the point-slope form. Here, the slope m is h'(3) = 12, and the point (x_1, y_1) is (3, h(3)) = (3, 3).
5. Step 5: Write the equation of the tangent line using the point-slope form: y - 3 = 12(x - 3).