Gilbert Strang Doubt

Question

Locate the inflection points and region where f(x) is concave up or down
f(x)=$-x^3+x^2+x$

Answer 1

f(x)=−x³+x²+x

f^'(x) = -3x² + 2x +1

f'' (X) = -6x+2

f'' (X) = 0

x = 1/3

we know that if f''(x) < 0 means it's maximum therefore it's concave down

similarly if f''(x) >0 means it's minimum therefore it's concave Up

f(x) is concave up (- $\infty$ ,1/3)

f(x) is concave down (1/3 , +  $\infty$ )

Point of inflection : f(1/3) = 11/27

therefore point of inflection : (1/3 , 11/27)