Question

P and W are twice-differentiable functions with P(7)=2(W(7)). The line y=9+2.5(x-7) is tangent to the graph of P at x=7. The line y=4.5+2(x-7) is tangento to the graph of WQ at x=7. Let m be a function such that m(x)=x(W(x)). Find m'(7).

Answer 1

Using the product rule, we have

$m(x) = x W(x) \implies m'(x) = xW'(x) + W(x)$

so that

$m'(7) = 7W'(7) + W(7)$

The equation of the tangent line to W(x) at x = 7 has all the information we need to determine m' (7).

When x = 7, the tangent line intersects with the graph of W(x), and

y = 4.5 + 2 (7 - 7)   ==>   y = 4.5

means that this intersection occurs at the point (7, 4.5), and this in turn means W (7) = 4.5.

The slope of this tangent line is 2, so W' (7) = 2.

Then

$m'(7) = 7\cdot2 + 4.5 = \boxed{18.5}$