Let f(x)= x x ;(x 1) Consider the following statements: 1. f^ (e)= 1 e 2. f ( x ) attains local minimum value at x=e 3. A local minimum value of f(x) is e Which of the statements given above are correct?

Step-by-step solution

aligned f^ (x) & = ( x)^2 1 x -( x-1) 2 x x ( x)^4 & = ( x)^2-2( x)^2 2 x x( x)^4 aligned f^ (e)= 1-2+2 e 1 = 1 e ( 0) Statement (1) is correct. Since f^ (e) 0 (true) f(x) attains local minima at x=e Statement (2) is correct. A local minimum value occurs at x=ef(x) at x=e , is f(e)= e e =e Statement (3) is correct. Hence, Statements (1), (2) and (3) are correct.