returns of the function. The point of diminishing returns is where the rate of yield per unit of input reaches its maximum and after which it will start to decrease. This is the point where the second derivative of the production function equals to zero.
First of all, we have our function P(t)=-5t^3+135t^2+6t. The first step is to find the first derivative of this function. We apply the power rule of derivatives (d/dx [x^n] = n*x^(n-1)) to each term of the function.
The derivative of -5t^3 is -15t^2 (bring the exponent in front and subtract the exponent by one).
The derivative of 135t^2 is 270t (bring the exponent in front and subtract the exponent by one).
The derivative of 6t is 6 (because the power of t is 1, its derivative is 1).
Adding these up, the first derivative of the function P(t) or P'(t) is -15t^2 + 270t + 6.
Next, we take the derivative of P'(t) to find the second derivative P''(t). Again, applying the power rule:
The derivative of -15t^2 is -30t.
The derivative of 270t is 270 (because the power of t is 1, its derivative is 1).
The derivative of the constant term 6 is 0.
So, adding these up, the second derivative P''(t) is -30t + 270.
We want to find when P''(t) is equal to 0. So we set P''(t) = 0, and we solve for t.
That gives us -30t + 270 = 0, which we can simplify to t = 9.
This tells us that the point of diminishing returns is at t=9.
To find the actual production value at this point, we substitute t = 9 back into our original function P(t) = -5t^3 + 135t^2 + 6t. This gives us -5*(9)^3 + 135*(9)^2 + 6*(9), which will provide the exact production value at the point of diminishing returns.
It is important to note that the point of diminishing returns does not necessarily mean the production is at its highest level, it just means production efficiency starts to decrease after this point.