Math / Calculus

Question12. [12 marks] Consider the function $y=-x^{4}+4 x^{3}$ . a. Find the $x$ -and $y$ -intercepts of the function. $x-i n t=(4,0), y-i n t:(0,0)$ b. Determine the intervals where the function is increasing/decreasing and find the relative extrema. c. Determine the intervals where the function is concave up/ concave down and find the inflection points. d. Sketch the function.

Studdy Solution
The x-intercepts are $(0, 0)$ and $(4, 0)$ , and the y-intercept is $(0, 0)$ .
The function is increasing on $(-\infty, 3)$ and decreasing on $(3, \infty)$ , with a relative maximum at $(3, 27)$ .
It's concave up on $(0, 2)$ and concave down on $(-\infty, 0)$ and $(2, \infty)$ , with inflection points at $(0, 0)$ and $(2, 16)$ .

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