Function f(x)=2x3−9x2+12x+29𝑓𝑥=2𝑥3-9𝑥2+12𝑥+29 is decreasing when x∈(a, b)𝑥∈𝑎, 𝑏 then find value of a+b𝑎+𝑏.

To find the interval where the function is decreasing, we first need to find the derivative of the function f(x) = 2x^3 - 9x^2 + 12x + 29.

The derivative of the function, f'(x), is 6x^2 - 18x + 12.

Next, we set the derivative equal to zero and solve for x to find the critical points:

6x^2 - 18x + 12 = 0

Divide the entire equation by 6 to simplify:

x^2 - 3x + 2 = 0

Factor the equation:

(x - 1)(x - 2) = 0

Setting each factor equal to zero gives the solutions x = 1 and x = 2. These are the critical points.

To determine where the function is decreasing, we test the intervals around the critical points in the derivative.

Choose a test point to the left of x = 1, between x = 1 and x = 2, and to the right of x = 2. Let's choose x = 0, x = 1.5, and x = 3.

Substitute x = 0 into the derivative: f'(0) = 12 > 0, so the function is increasing on the interval (-∞, 1).

Substitute x = 1.5 into the derivative: f'(1.5) = -1.5 < 0, so the function is decreasing on the interval (1, 2).

Substitute x = 3 into the derivative: f'(3) = 9 > 0, so the function is increasing on the interval (2, ∞).

Therefore, the function is decreasing when x ∈ (1, 2). The sum of a + b is 1 + 2 = 3.