answer:The factored form of the expression is 11(7-x)(x-1).

answer:The product of p(x) and q(x), after expanding, is: left(frac{27x^2}{2} + frac{27x}{4} - frac{23}{4}right) left(frac{45x^2}{4} - frac{45x}{4} + frac{41}{4}right) = frac{1215x^4}{8} - frac{1215x^3}{16} - frac{9x^2}{4} + frac{1071x}{8} - frac{943}{16}

answer:Let's dive right in. We have {eq}begin{align*} frac{d}{dx} f(x) &= lim_{Delta x to 0} frac{f(x+Delta x)-f(x)}{Delta x} &= lim_{Delta x to 0} frac{ 4/sqrt{x+Delta x} - 4/ sqrt x}{Delta x} &= lim_{Delta x to 0} frac{ 4sqrt x - 4 sqrt{x+Delta x}}{Delta x sqrt{x+Delta x} sqrt x} end{align*} {/eq} Now, we can multiply by what is essentially the conjugate of the numerator over itself to write {eq}begin{align*} frac{d}{dx} f(x) &= lim_{Delta x to 0} frac{ 4sqrt x - 4 sqrt{x+Delta x}}{Delta x sqrt{x+Delta x} sqrt x} cdot frac{4sqrt x + 4 sqrt{x+Delta x}}{4sqrt x + 4 sqrt{x+Delta x}} &= lim_{Delta x to 0} frac{ 16(x-(x+Delta x))}{(Delta x sqrt{x+Delta x} sqrt x)(4sqrt x + 4 sqrt{x+Delta x})} &= lim_{Delta x to 0} frac{ -16Delta x}{4(Delta x sqrt{x+Delta x} sqrt x)(sqrt x + sqrt{x+Delta x})} &= lim_{Delta x to 0} frac{ -4}{(sqrt{x+Delta x} sqrt x)(sqrt x + sqrt{x+Delta x})} &= -frac{ 4}{(sqrt{x} sqrt x)(sqrt x + sqrt{x})} &= -frac4{2xsqrt x} &= - frac2{x sqrt x} end{align*} {/eq} Therefore, the derivative of {eq}f(x) = frac 4 {sqrt x } {/eq} is {eq}f'(x) = - frac2{x sqrt x} {/eq}.

answer:This is a linear programming problem. The optimal solution is M_1 = sqrt{P} and M_i = 0 for all i > 1. This solution satisfies the constraints and maximizes the objective function because lambda_i are non-negative and in decreasing order.