急~f4amaths,,,,今日要交啦

用戶209328

2009-02-15 10:58 pm

1.if @takes any real values,find the range oof the values of y
y=1/2sin^2@+5cos^2@

2.consider f@=11-6cos@-3sin^2@
find the max.& mini.values of f@ if @ takes any real values

回答 (1)

老爺子

2009-02-15 11:57 pm

✔ 最佳答案

1.
y = (1/2)sin2@ + 5cos2@
y = (1/2)sin2@ + (1/2)cos2@ + (9/2)cos2@
y = (1/2)(sin2@ + cos2@) + (9/2)cos2@
y = (1/2) + (9/2)cos2@

-1 ≤ cos@ ≤ 1
0 ≤ cos2@ ≤ 1
(1/2) + 0 ≤ (1/2) + (9/2)cos2@ ≤ (1/2) + (9/2)(1)
0 ≤ y ≤ 5

2.
f@ = 11 - 6cos@ - 3sin2@
f@ = 11 - 6cos@ - 3(1 - cos2@)
f@ = 11 - 6cos@ - 3 + 3cos2@
f@ = 3cos2@ - 6cos@ + 8
f@ = 3(cos2@ - 2cos@ + 1) - 3 + 8
f@ = 3(cos@ - 1)2 + 5

-1 ≤ cos@ ≤ 1
-2 ≤ cos@ - 1 ≤ 0
0 ≤ (cos@ - 1)2 ≤ 4
3(0) + 5 ≤ 3(cos@ - 1)2 + 5 ≤ 3(4) + 5
5 ≤ f@ ≤ 17

The maximum value of f@ is 17, and the minimum value is 5.
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