Let $f$ and $g$ be differentiable functions for which the following information is known: $f(2)=5,g(2)=-3,f^{\text{'}}(2)=-\frac{1}{2},g^{\text{'}}(2)=2.$

a$.$ Let $h$ be the new function defined by the rule $h(x)=3f(x)-4g(x).$ Determine $h(2)$ and $h^{\text{'}}(2).$

b$.$ Find an equation for the tangent line to $y=h(x)$ at the point $(2,h(2)).$

c$.$ Let $p$ be the function defined by the rule $p(x)=-2f(x)\frac{1}{2}g(x).$ Is $p$ increasing, decreasing, or neither at $a=2?$ Why?

d$.$ Estimate the value of $p(2\mathrm{.}03)$ by using the local linearization of $p$ at the point $(2,p(2)).$