1. The problem asks: When does a tangent line approximation underestimate the value of a function? 2. The tangent line approximation at a point $x=a$ uses the linearization formula: $$L(x) = f(a) + f'(a)(x - a)$$ This line approximates $f(x)$ near $x=a$. 3. Important rule: If the function $f$ is concave up (i.e., $f''(x) > 0$) near $a$, the tangent line lies below the curve, so the tangent line approximation underestimates $f(x)$. 4. Conversely, if $f$ is concave down ($f''(x) < 0$), the tangent line lies above the curve, so the approximation overestimates $f(x)$. 5. Therefore, the guarantee for an underestimate using tangent line approximation is that the function is concave up at the point of tangency. 6. In summary: - Tangent line approximation underestimates $\iff f''(a) > 0$ (function is concave up at $a$). Final answer: The estimate using a tangent line approximation will result in an underestimate if the function is concave up at the point of tangency.