First slide

Application of diffrentiation

Question

Find maximum or minimum values of the functions
$y = 9 - (x - 3)^{2}$

Moderate

A

3
B

6
C

18
D

9

Solution

$y = 9 - (x - 3)^{2} = 9 - x^{2} - 9 + 6 x$
$y = 6 x - x^{2}$
$\frac{d y}{d x} = 6 - 2 x$
For minimum or maximum value of y. We will substitute $\frac{d y}{d x} = 0$
6 - 2x=0 or x=3
To check whether value of y is maximum or minimum at x=3 we will have to check whether $\frac{d^{2} y}{d x^{2}}$ is positive or negative.
$\frac{d^{2} y}{d x^{2}} = - 2$
or $\frac{d^{2} y}{d x^{2}}$ is negative at x=3. Hence, value of y is maximum. This maximum value of y is
$y_{\max} = 9 - (3 - 3)^{2} = 9$

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