Let domain and range of f(x) and g(x) are respectively [ 0 , ∞ ) . If f ( x ) be an increasing function and g(x) be an decreasing function. Also, h ( x ) = f ( g ( x )) , h ( 0 ) = 0 and p ( x ) = h (

Question

Let domain and range of f(x) and g(x) are respectively [ 0 , ∞ ) . If f ( x ) be an increasing function and g(x) be an decreasing function. Also, h ( x ) = f ( g ( x )) , h ( 0 ) = 0 and p ( x ) = h ( x 3 − 2 x 2 + 2 x ) − h ( 4 ) then for every x ∈ ( 0 , 2 ]

TestHub · Accepted Answer

h ( x ) = f ( g ( x )) h ′ ( x ) = f ′ ( g ( x )) g ′ ( x ) < 0 ∀ x ∈ [ 0 , ∞ ) because g ′ ( x ) < 0 ∀ x ∈ [ 0 , ∞ ) and f ′ ( g ( x )) > 0 ∀ x ∈ [ 0 , ∞ ) . Also, h ( 0 ) = 0 and hence, h ( x ) < 0 ∀ x ∈ [ 0 , ∞ ) . p ( x ) = h ( x 3 − 2 x 2 + 2 x ) − h ( 4 ) p ′ ( x ) = h ′ ( x 3 − 2 x 2 + 2 x ) ⋅ ( 3 x 2 − 4 x + 2 ) < 0 ∀ x ∈ ( 0 , 2 ) Since 3 x 2 − 4 x + 2 > 0 ∀ x ∈ R , p ( x ) is a decreasing function. This implies p ( 2 ) < p ( x ) < p ( 0 ) ∀ x ∈ ( 0 , 2 ) . So, h ( 4 ) − h ( 4 ) < p ( x ) < h ( 0 ) − h ( 4 ) . Therefore, 0 < p ( x ) < − h ( 4 ) .

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