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Sufficient Condition for an Extrema

Find maximum ...

Question

Find maximum value of y in $y = 5 - {(x - 1)}^{2}$

y_{m a x} = 5 a t x = 1

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y_{m a x} = 15 a t x = 1

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y_{m a x} = 3 a t x = 1

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None of these

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Solution

The correct option is B $y_{m a x} = 5 a t x = 1$
Given, $y = 5 - (x - 1)^{2}$

For maximum or minimum of $y$ , $\frac{d y}{d x} = 0$

So, $- 2 (x - 1) = 0$ or $x = 1$

Thus, $\frac{d^{2} y}{d x^{2}} = - 2 < 0$

Hence, at $x = 1$ the function $y$ will have maximum value.

Thus, $y_{m a x} = 5 - (1 - 1)^{2} = 5$

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