What is the extreme 4sin(x)+4cos(x),0<= x<= pi/2 ?

The extreme 4sin(x)+4cos(x),0<= x<= pi/2 is Minimum(0,4),Maximum(pi/4 ,4sqrt(2)),Minimum(pi/2 ,4)

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extreme 4sin(x)+4cos(x),0<= x<= pi/2

extreme points

4 sin (x) + 4 cos (x), 0 \leq x \leq \frac{π}{2}

Minimum (0, 4), Maximum (\frac{π}{4}, 42), Minimum (\frac{π}{2}, 4)

Solution steps

Find the critical points:

x = 0, x = \frac{π}{4}, x = \frac{π}{2}

Domain of

4 sin (x) + 4 cos (x), 0 \leq x \leq \frac{π}{2} : 0 \leq x \leq \frac{π}{2}

Find intervals:

Increasing

: 0 < x < \frac{π}{4},

Decreasing

: \frac{π}{4} < x < \frac{π}{2}

Plug the extreme point

x = 0

into

4 sin (x) + 4 cos (x) \Rightarrow y = 4

Minimum (0, 4)

Plug the extreme point

x = \frac{π}{4}

into

4 sin (x) + 4 cos (x) \Rightarrow y = 42

Maximum (\frac{π}{4}, 42)

Plug the extreme point

x = \frac{π}{2}

into

4 sin (x) + 4 cos (x) \Rightarrow y = 4

Minimum (\frac{π}{2}, 4)

Minimum (0, 4), Maximum (\frac{π}{4}, 42), Minimum (\frac{π}{2}, 4)

f (x, y) = x^{2} - x y + y^{2} - 4 x + 4 y

f (x, y) = e^{2 y} (2 x^{2} - 2 x y + 1)

f (x) = 2 x^{2} - 4 x + y^{2} - 6 y + 1

f (x) = x^{3} - 12 x^{2} - 27 x + 2, - 2 \leq x \leq 0

f (x) = - 4 x^{3} + 12 x^{2}

What is the extreme 4sin(x)+4cos(x),0<= x<= pi/2 ?
The extreme 4sin(x)+4cos(x),0<= x<= pi/2 is Minimum(0,4),Maximum(pi/4 ,4sqrt(2)),Minimum(pi/2 ,4)