The diagram shows the graph of y=4-3x^2, for x≥0 and y≥0. B is a point on the graph and OABC is a rectangle. Find the value of x for which the area of OABC is a maximum.
+26367 Applications of differentiation and antidifferentiation
The diagram shows the graph of y=4-3x^2, for x=0 and y=0.
B is a point on the graph and OABC is a rectangle.
Find the value of x for which the area of OABC is a maximum.
...here is no diagram.
I assume:
\(\text{Let $\mathbf{A}$ is the area of OABC }\)
\(\begin{array}{|rcll|} \hline A &=& x\cdot y \\ &=& x\cdot (4-3x^2) \\ &=& 4x-3x^3 \\\\ A' &=& 4-9x^2 \quad | \quad A' = 0 \\ 4-9x^2 &=& 0 \\ 9x^2 &=& 4 \quad | \quad \text{square root both sides} \\ 3x &=& 2 \\\\ \mathbf{x} &=& \mathbf{\dfrac{2}{3}} \quad | \quad A''=-18x =-18\cdot \dfrac{2}{3}=-12\ (A''<0 \text{ maximum}) \\ \hline \end{array}\)
