Higher Order Derivatives — Question 1

Question 1 - Solution

We are given: $$f(x) = x^4 - 4x^3 + 6x^2$$

(a) First derivative: $$f'(x) = 4x^3 - 12x^2 + 12x$$

Second derivative: $$f''(x) = 12x^2 - 24x + 12$$

Factor: $$f''(x) = 12(x^2 - 2x + 1) = 12(x - 1)^2$$

Answer: $$f'(x) = 4x^3 - 12x^2 + 12x, \quad f''(x) = 12(x - 1)^2$$

(b) Analyze sign of $f''(x)$:

Since $f''(x) = 12(x - 1)^2 \geq 0$ for all $x$, the second derivative is never negative.

$f''(x) > 0$ for $x \ne 1$ → concave up

$f''(1) = 0$ → possible inflection point

But since the sign does not change around $x = 1$, it is not an inflection point.

Answer: Concave up: $(-\infty, \infty)$ Concave down: $\text{None}$

(c) No change in concavity at $x = 1$, so:

Answer: $\boxed{\text{No points of inflection}}$