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# Applications of differentiation and antidifferentiation

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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.

Aug 28, 2019

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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:

$$\text{Let \mathbf{A} is the area of OABC }$$

$$\begin{array}{|rcll|} \hline A &=& x\cdot y \\ &=& x\cdot (4-3x^2) \\ &=& 4x-3x^3 \\\\ A' &=& 4-9x^2 \quad | \quad A' = 0 \\ 4-9x^2 &=& 0 \\ 9x^2 &=& 4 \quad | \quad \text{square root both sides} \\ 3x &=& 2 \\\\ \mathbf{x} &=& \mathbf{\dfrac{2}{3}} \quad | \quad A''=-18x =-18\cdot \dfrac{2}{3}=-12\ (A''<0 \text{ maximum}) \\ \hline \end{array}$$ Aug 28, 2019
edited by heureka  Aug 28, 2019
edited by heureka  Aug 28, 2019
edited by heureka  Aug 28, 2019
edited by heureka  Aug 29, 2019
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Thanks, heureka.....

Here is the graph  of the max rectangle :    Aug 28, 2019
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Thank you, CPhill ! heureka  Aug 29, 2019