Applications of Derivatives
Minimum area of triangle OAB from tangent line
MJAT_TS8_P2
Grade 12
Question:
Let $f(x)=5-(x+1)^2$. The tangent at $P(h,f(h))$ meets the $x$-axis at $A$ and $y$-axis at $B$. Let $S(x)$ = area of $\triangle OAB$. The minimum value of $S(x)$ in $(0,1)$ is $p/q$ where $p,q$ are the numerically smallest possible integers. Find $10q^2-p$:
Step-by-Step Solution
Key Concept: Tangent at $(h,k)$ where $k=5-(h+1)^2$: $y-k=-2(h+1)(x-h)$. $A=(h+k/(2(h+1)),0)$ and $B=(0,k+2h(h+1))$. Area $S=\frac{1}{2}|OA||OB|$.
$10q^2-p=\mathbf{95}$.
Correct Answer: 95