Question

question 10
find the inflection point(s) for the function $f(x) = -x^5 + 5x^4 + 12x - 2$.
(0, -2) and (3,196)
(3,196)
(3,0)
(3,147)

Explanation:

Step1: Compute first derivative

$f'(x) = \frac{d}{dx}(-x^5 + 5x^4 + 12x - 2) = -5x^4 + 20x^3 + 12$

Step2: Compute second derivative

$f''(x) = \frac{d}{dx}(-5x^4 + 20x^3 + 12) = -20x^3 + 60x^2$

Step3: Solve $f''(x)=0$

$-20x^3 + 60x^2 = 0$
$-20x^2(x - 3) = 0$
Solutions: $x=0$, $x=3$

Step4: Test concavity at $x=0$

For $x=-1$: $f''(-1) = -20(-1)^3 + 60(-1)^2 = 80 > 0$ (concave up)
For $x=1$: $f''(1) = -20(1)^3 + 60(1)^2 = 40 > 0$ (concave up)
No concavity change at $x=0$, so it is not an inflection point.

Step5: Test concavity at $x=3$

For $x=2$: $f''(2) = -20(2)^3 + 60(2)^2 = 80 > 0$ (concave up)
For $x=4$: $f''(4) = -20(4)^3 + 60(4)^2 = -320 < 0$ (concave down)
Concavity changes at $x=3$, so it is an inflection point.

Step6: Find $f(3)$

$f(3) = -(3)^5 + 5(3)^4 + 12(3) - 2 = -243 + 405 + 36 - 2 = 196$

Answer:

(3,196)