Question

suppose that ( h(x)=f(x)-2g(x), f(-1)=3 ) and ( g(-1)=-2 ).
find ( h(-1) ).

Explanation:

Step1: Differentiate \( h(x) \)

According to the sum - difference rule of differentiation \((u - v)^\prime=u^\prime - v^\prime\). If \(h(x)=f(x)-2g(x)\), then \(h^\prime(x)=f^\prime(x)-2g^\prime(x)\) (by the constant multiple rule \((cu)^\prime = cu^\prime\) where \(c = 2\)).

Step2: Substitute \(x=-1\)

We know that \(f^\prime(-1) = 3\) and \(g^\prime(-1)=-2\). Substitute these values into the formula for \(h^\prime(x)\) at \(x =-1\). So \(h^\prime(-1)=f^\prime(-1)-2g^\prime(-1)\).
Substitute \(f^\prime(-1) = 3\) and \(g^\prime(-1)=-2\) into the right - hand side: \(h^\prime(-1)=3-2\times(-2)\).
First, calculate \(2\times(-2)=-4\). Then \(h^\prime(-1)=3+4\).

Answer:

\(7\)