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Consider the function h(x)=(g^(2)(x))/(2...

Consider the function `h(x)=(g^(2)(x))/(2)+3x^(3)-5`, where g(x) is a continuous and differentiable function. It is given that h(x) is a monotonically increasing function and g(0) = 4. Then which of the following is not true ?

A

`g^(2)(1)gt10`

B

`h(5)gt3`

C

`h((5)/(2))lt2`

D

`g^(-1)lt22`

Text Solution

Verified by Experts

The correct Answer is:
C
`h(x)=(g^(2)(x))/(2)+3x^(3)-5`
`h'(x)gt0`
`rArr" "g(x)g'(x)gt-9x^(2)`
`rArr" "int_(0)^(1)g(x)g'(x)dxgt-int_(0)^(1)9x^(2)dx`
`rArr" "((g(1))^(2)-(g(x))^(2))/(2)gt-3(1-0)`
`rArr" "(g(1))^(2)-16gt-6`
`rArr" "(g(1))^(2)gt10`
`" "int_(-1)^(0)g(x)g'(x)dx gt -int_(-1)^(0)9x^(2)dx`
`rArr" "((g(0))^(2)-(g(-1))^(2))/(2)gt-3(0-(-1))`
`rArr" "16-(g(-1))^(2)gt-6`
`rArr" "(g(-1))^(2)lt22`
`" "h(5)gth(0)`
`rArr" "h(5)gt(g^(2)(0))/(2)+3(0)-5=3`
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