Question

# Let f(x)=1(cosx−3)2+(sinx+4)2 and g(x)=√3sinx+cosx. Then which of the following is/are true ?

A
Maximum value of f(x) is 136
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B
Minimum value of f(x) is 136
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C
Maximum value of g(x) occurs at x=π3
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D
Sum of the minimum values of f(x) and g(x) is 7136
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Solution

## The correct options are B Minimum value of f(x) is 136 C Maximum value of g(x) occurs at x=π3 D Sum of the minimum values of f(x) and g(x) is −7136 f(x)=1(cosx−3)2+(sinx+4)2 =1cos2x−6cosx+9+sin2x+8sinx+16 =18sinx−6cosx+26 Extreme values of 8sinx−6cosx+26 are 26±√(8)2+(−6)2=26±10=36,16 ∴fmax=116 and fmin=136 g(x)=√3sinx+cosx=2(√32sinx+12cosx) g(x)=2sin(x+π6) ∴gmax occurs at x=π3 and gmin=−2 Sum of minimum values of f(x) and g(x) is 136+(−2)=−7136

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