Select the correct function type for function f(x) on (1,3): (decreasing function, convex function or decreasing function, concave function or increasing function, convex function or increasing function, concave function or non of the others) f(x)=(x+1)(x-10) f(x)=\frac{1}{3}x^3-4x^2+3x-1 f(x)=\frac{1}{2}x^2-2 f(x)=-x(x-8)

To determine the type of function, we need to look at the first and second derivatives of the function. The first derivative tells us whether the function is increasing or decreasing, while the second derivative tells us whether the function is concave or convex.

1. f(x)=(x+1)(x-10) The first derivative is f'(x) = 2x - 9. This is positive for x > 4.5 and negative for x < 4.5. So the function is increasing on (1,3). The second derivative is f''(x) = 2, which is always positive. So the function is convex. Therefore, the function is an increasing, convex function on (1,3).

2. f(x)=\frac{1}{3}x^3-4x^2+3x-1 The first derivative is f'(x) = x^2 - 8x + 3. This is positive for x > 4 and negative for x < 4. So the function is decreasing on (1,3). The second derivative is f''(x) = 2x - 8, which is negative for x < 4 and positive for x > 4. So the function is concave on (1,3). Therefore, the function is a decreasing, concave function on (1,3).

3. f(x)=\frac{1}{2}x^2-2 The first derivative is f'(x) = x. This is positive for x > 0 and negative for x < 0. So the function is increasing on (1,3). The second derivative is f''(x) = 1, which is always positive. So the function is convex. Therefore, the function is an increasing, convex function on (1,3).

4. f(x)=-x(x-8) The first derivative is f'(x) = -2x + 8. This is positive for x < 4 and negative for x > 4. So the function is decreasing on (1,3). The second derivative is f''(x) = -2, which is always negative. So the function is concave. Therefore, the function is a decreasing, concave function on (1,3).