Determine the concavity and find the inflection point of the function | Basic Mathematics 2077

Basic Mathematics20775 marksNumerical: concavity and inflection pointConcavity and inflection points

Determine the concavity and find the inflection point of the function $f(x) = x^3 - 3x^2 + 2$ . [5]

Determine the concavity and find the inflection point of the function $f(x) = x^3 - 3x^2 + 2$. [5]

Function: $f(x) = x^3 - 3x^2 + 2$ $f'(x) = 3x^2 - 6x$ $f''(x) = 6x - 6$ Set $f''(x) = 0$ : $6x - 6 = 0 \implies x = 1$ - For $x < 1$ (test $x = 0$ ): $f''(0) = 6(0) - 6 = -6 < 0$ → concave down on $(-\infty, 1)$ - For $x 1$ (test $x = 2$ ): $f''(2) = 6(2) - 6 = 6 0$ → concave up on

Function: $f(x) = x^3 - 3x^2 + 2$ $$f'(x) = 3x^2 - 6x$$ $$f''(x) = 6x - 6$$ Set $f''(x) = 0$: $$6x - 6 = 0 \implies x = 1$$ - For $x < 1$ (test $x = 0$): $f''(0) = 6(0) - 6 = -6 < 0$ → concave down on $(-\infty, 1)$ - For $x 1$ (test $x = 2$): $f''(2) = 6(2) - 6 = 6 0$ → concave up on

Determine the concavity and find the inflection point of the function f(x) = x3 - 3x2 + 2.