Question

Show that if \(F ^ { \prime } ( x ) = G ^ { \prime } ( x )\) on \([ a , b ] ,\) then \(F ( b ) - F ( a ) = G ( b ) - G ( a )\)

Short answer

Proof relies on the properties of derivatives and the constant function rule. If two functions have the same derivative over some interval, then the difference in their values at the endpoints of this interval is the same. So, the statement \(F ( b ) - F ( a ) = G ( b ) - G ( a )\) holds true.

Step-by-step solution

Assume the hypothesis

Assume that \(F ^ { \prime } ( x ) = G ^ { \prime } ( x )\) for all \(x\) in \([ a , b ] .\)

Define a new function

Define \(H ( x ) = F ( x ) - G ( x )\) for all \(x\) in \([ a , b ]\). Since \(F\) and \(G\) are differentiable on \( [ a , b ] \), so is the function \(H(x)\).

Calculate the derivative of the new function

By properties of derivatives, \(H ^ { \prime } ( x ) = F ^ { \prime } ( x ) - G ^ { \prime } ( x )\). But since \(F ^ { \prime } ( x ) = G ^ { \prime } ( x )\) for all \(x\) in \([ a , b ]\), then \(H ^ { \prime } ( x ) = 0\) for all \(x\) in \([ a , b ]\).

Conclude the result

If a function has zero derivative on an interval, it must be constant on that interval (according to the constant function rule in calculus). Therefore, \(H(x) = c\), for some constant \(c\). Since \(H(a) = H(b)\), we have \(F(a) - G(a) = F(b) - G(b)\). Thus, \(F ( b ) - F ( a ) = G ( b ) - G ( a )\).