Question: Let $f \left( x ^ { 2 } \right) = x ^ { 2 } ( 1 + x )$ then f (4) equals...

Question

Let $f \left( x ^ { 2 } \right) = x ^ { 2 } ( 1 + x )$ then f (4) equals

Answer 1

By definition of f(x) we have $f \left( x ^ { 2 } \right) = \int _ { 0 } ^ { x ^ { 2 } } f ( t ) d t = x ^ { 2 } + x ^ { 3 }$

(given)

Differentiate both sides, $f \left( x ^ { 2 } \right) \cdot 2 x + 0 = 2 x + 3 x ^ { 2 }$

Put, $x = 2 \Rightarrow 4 f ( 4 ) = 16 \Rightarrow f ( 4 ) = 4$

Question: Let \(f \left( x ^ { 2 } \right) = x ^ { 2 } ( 1 + x )\) then f (4) equals...