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Popular >

extreme f(x)=sin(x),0<= x<pi^2

Solution

extreme points

f (x) = sin (x), 0 \leq x < π^{2}

Solution

Minimum (0, 0), Maximum (\frac{π}{2}, 1), Minimum (\frac{3 π}{2}, - 1), Maximum (\frac{5 π}{2}, 1)

Solution steps

Find the critical points:

x = 0, x = \frac{π}{2}, x = \frac{3 π}{2}, x = \frac{5 π}{2}

Domain of

sin (x), 0 \leq x < π^{2} : 0 \leq x < π^{2}

Find intervals:

Increasing

: 0 < x < \frac{π}{2},

Decreasing

: \frac{π}{2} < x < \frac{3 π}{2},

Increasing

: \frac{3 π}{2} < x < \frac{5 π}{2},

Decreasing

: \frac{5 π}{2} < x < π^{2}

Plug the extreme point

x = 0

into

sin (x) \Rightarrow y = 0

Minimum (0, 0)

Plug the extreme point

x = \frac{π}{2}

into

sin (x) \Rightarrow y = 1

Maximum (\frac{π}{2}, 1)

Plug the extreme point

x = \frac{3 π}{2}

into

sin (x) \Rightarrow y = - 1

Minimum (\frac{3 π}{2}, - 1)

Plug the extreme point

x = \frac{5 π}{2}

into

sin (x) \Rightarrow y = 1

Maximum (\frac{5 π}{2}, 1)

Minimum (0, 0), Maximum (\frac{π}{2}, 1), Minimum (\frac{3 π}{2}, - 1), Maximum (\frac{5 π}{2}, 1)

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