Maximum and Minimum value of f ( x ) f(x) .

Geometry Level 2

If f ( x ) = ( sin ( x ) + cos ( x ) ) 2 f(x) = (\sin(x) + \cos(x))^2 . Find out the minimum and maximum value of f ( x ) f(x) .

0, 2 1, 2 1, 3 -1, 0

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2 solutions

Chew-Seong Cheong
Dec 20, 2017

f ( x ) = ( sin ( x ) + cos ( x ) ) 2 = ( 2 sin ( x + π 4 ) ) 2 = 2 sin 2 ( x + π 4 ) Since 0 sin 2 ( x + π 4 ) 1 0 f ( x ) 2 \begin{aligned} f(x) & = (\sin(x) + \cos(x))^2 \\ & = \left(\sqrt 2 \sin \left(x+\frac \pi 4\right)\right)^2 \\ & = 2 \color{#3D99F6} \sin^2 \left(x+\frac \pi 4\right) & \small \color{#3D99F6} \text{Since }0 \le \sin^2 \left(x+\frac \pi 4\right) \le 1 \\ \implies 0 & \le f(x) \le 2 \end{aligned}

Srinivasa Gopal
Dec 18, 2017

F(X) = (Sin(X) + Cos(X) )^2 Expanding the expression F(X) = Sin(X)^2 + Cos(X)^2 + 2* Sin(X) * Cos(X)

F(X) = 1 +Sin(2X)

Easy to see that the Min value of F(X) occurs when Sin(2X) = -1 and maximum value occurs when Sin(2X) = 1.

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