InterviewSolution
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Consider the unction f(x)=int_(0)^(x)(5ln(1+t^(2))-10t tan^(-1)t+16sint)dt f(x) is |
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Answer» negative for all `X in (0,1)` `rArr""f'(x)=5ln(1+x^(2))-10x tan^(-1)x+16 sinx` `rArr""f''(x)=2(8 cos x-5 tan^(-1)x)` `rArr""f''(x)=-2(8sinx+(5)/(1+x^(2)))lt0AAx in (0,1)` So, f''(x) is decreasing `AA x in (0,1)` `rArr""f''(x)gtf''(1)=2(8cos1-(5pi)/(4))` `""gt2(8COS.(pi)/(3)-(5pi)/(4))` `""=2(4-(5pi)/(4))gt0` So, f''(x) is increasing, for `x gt 0 , f'(x)gtf'(0)=0` So, f(x) is increasing, for `x gt0, f(x) gt f(0)=0` So, `int_(0)^(x)f(t)` is positive and increasing. |
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