find-the-minimum-and-maximum-value-of-the-quadratic-functions-a-4x-2-5x-1-b-x-2-x-3-c-x-2-x-4-6-hence-draw-each-draw-

Question Number 39607 by Rio Mike last updated on 08/Jul/18

f i n d t h e

m i n i m u m a n d m a x i m u m v a l u e

o f t h e q u a d r a t i c f u n c t i o n s

a) 4 x^{2} + 5 x + 1

b) x + \frac{2}{x} = 3

c) x^{2} - \frac{x}{4} + 6

h e n c e d r a w e a c h d r a w

Answered by tanmay.chaudhury50@gmail.com last updated on 08/Jul/18

a) 4 x^{2} + 5 x + 1

4 (x^{2} + \frac{5}{4} x + \frac{1}{4})

4 {(x^{2} + 2 . x . \frac{5}{8} + \frac{25}{64} + \frac{1}{4} - \frac{25}{64})}

4 {{(x + \frac{5}{8})}^{2} + \frac{16 - 25}{64}}

4 {{(x + \frac{5}{8})}^{2} + \frac{- 9}{64}}

4 {(x + \frac{5}{8})}^{2} - \frac{9}{16}

{(x + \frac{5}{8})}^{2} > 0

a s t h e v a l u e o f x i n c r e a s e s s o t h e v a l u e o f

{(x + \frac{5}{8})}^{2} i n c r e a s e s \dots h e n c e 4 x^{2} + 5 x + 1 h a s n o

m a x i m u m v a l u e . . i t s m i n i m u m v a l u e i s

\frac{- 9}{16} w h e n x = \frac{- 5}{8}

o t h e r m e t h o d \dots

b y u s i n g c a l c u l u s

y = 4 x^{2} + 5 x + 1

\frac{d y}{d x} = 8 x + 5

f o r m i n / m a x \frac{d y}{d x} = 0 s o x = - \frac{5}{8}

\frac{d^{2} y}{{d x}^{2}} = 8 t h a t m e a n s \frac{d^{2} y}{{d x}^{2}} > 0

s o 4 x^{2} + 5 x + 1 h a s m i n i m u m v a l u e a t x = - \frac{5}{8}

4 . (\frac{25}{64}) + 5 (\frac{- 5}{8}) + 1

\frac{25}{16} - \frac{25}{8} + 1

\frac{25 - 50 + 16}{16} = \frac{- 9}{16}

Answered by tanmay.chaudhury50@gmail.com last updated on 08/Jul/18

c) y = \frac{4 x^{2} - x + 24}{4}

\frac{1}{4} {{(2 x)}^{2} - 2.2 x . \frac{1}{4} + \frac{1}{16} + 24 - \frac{1}{16}}

\frac{1}{4} {{(2 x - \frac{1}{4})}^{2} + \frac{24 \times 16 - 1}{16}}

a s t h e v a l u e o f x i n c r e a s e s s o t h e v a l u e o f

e x p r e s s i o n i n c r e a s e s . . s o t h e e x p r e s s i o n h a s

n o m a x i m u m v a l u e . .

i t s m i n i m u m v a l u e i s \frac{24 \times 16 - 1}{4 \times 16} w h e n x = \frac{1}{8}

m i n v a l u e = \frac{383}{64} w h e n x = \frac{1}{8}

b y c a l c u l u s

y = x^{2} - \frac{x}{4} + 6

\frac{d y}{d x} = 2 x - \frac{1}{4}

f o r m i n / m a x \frac{d y}{d x} = 0 s o x = \frac{1}{8}

\frac{d^{2} y}{{d x}^{2}} = 2 s o \frac{d^{2} y}{{d x}^{2}} > 0

s o t h e e x p r e s s i o n h a s m i n v a l u e a t x = \frac{1}{8}

m i n v a l u d {(\frac{1}{8})}^{2} - \frac{1}{32} + 6

\frac{1 - 2 + 384}{64} = \frac{383}{64}

Answered by MJS last updated on 08/Jul/18

y = {a x}^{2} + b x + c

z e r o s a t x = - \frac{b}{2 a} \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a}

a > 0 \Rightarrow m i n a t x = - \frac{b}{2 a}; y = \frac{- b^{2} + 4 a c}{4 a^{2}}

a < 0 \Rightarrow m a x a t x = - \frac{b}{2 a}; y = \frac{- b^{2} + 4 a c}{4 a^{2}}

y = x^{2} + p x + q \Rightarrow m i n a t x = - \frac{p}{2}; y = \frac{- p^{2} + 4 q}{4}

z e r o s a t x = - \frac{p}{2} \pm \frac{\sqrt{p^{2} - 4 q}}{2}

y = - x^{2} + p x + q \Rightarrow m a x a t x = \frac{p}{2}; y = \frac{p^{2} + 4 q}{4}

z e r o s a t x = \frac{p}{2} \pm \frac{\sqrt{p^{2} + 4 q}}{2}

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