Find the inflection point(s) for the function $f(x)=-2 x^{6}+3 x^{5}+12 x-2$
$(-1,-19)$ and $(0,-2)$
$(0,12)$ and $(1,15)$
$(0,0)$ and $(1,0)$
$(0,-2)$ and $(1,11)$

Step 1 ：Given the function \(f(x)=-2 x^{6}+3 x^{5}+12 x-2\), we need to find the inflection point(s).

Step 2 ：The inflection point of a function is the point where the function changes concavity. In other words, it's where the second derivative of the function changes sign.

Step 3 ：To find the inflection points, we first need to find the second derivative of the function, set it equal to zero, and solve for x.

Step 4 ：The second derivative of the function \(f(x)=-2 x^{6}+3 x^{5}+12 x-2\) is \(f''(x) = 60x^{3}(1 - x)\).

Step 5 ：Setting \(f''(x)\) equal to zero gives us \(60x^{3}(1 - x) = 0\). Solving for x, we get the solutions x = 0 and x = 1.

Step 6 ：We then substitute these x-values into the original function to get the corresponding y-values. For x = 0, we get y = -2. For x = 1, we get y = 11.

Step 7 ：Thus, the inflection points for the function \(f(x)=-2 x^{6}+3 x^{5}+12 x-2\) are \((0,-2)\) and \((1,11)\).

Step 8 ：\(\boxed{\text{Final Answer: The inflection points for the function } f(x)=-2 x^{6}+3 x^{5}+12 x-2 \text{ are } (0,-2) \text{ and } (1,11)}\)