A certain quadratic function is defined as , where A and B are fixed constants not identified. Assume, however, that the equation of the tangent line to f(x), when x = 1, is . Calculate the exact values of A and B, if you know that the Mean Value Theorem is satisfied on the interval [0,2] when x = 1.

Difficulty:

Solution:

If the equation of the tangent line (when x = 1) is , then the slope of the tangent line (i.e. the derivative) is 3. Furthermore, both f and the tangent line must share the same coordinates at the point of tangency. Hence, we know that f(1) = 7, since 3(1) + 4 = 7. If you plug (1,7) into the equation, you get this: 2 + A + B = 7, or A + B = 5. In and of itself, this doesn't tell you a whole lot. However, let's apply the Mean Value Theorem, as instructed by the problem:

Since we already know that A + B = 5, then B = 6.