Step 1: Define a new function
Let us define a new function h(x) which represents the difference between f(x) and g(x):
h(x)=f(x)−g(x)
Step 2: Use the second derivative relation
We are given that:
f′′(x)=g′′(x)⟹f′′(x)−g′′(x)=0
Taking the second derivative of our assumed function h(x):
h′′(x)=f′′(x)−g′′(x)=0
Step 3: Integrate to find the first derivative
Integrating h′′(x)=0 with respect to x, we get:
h′(x)=C1
(where C1 is a constant of integration)
Since h′(x)=f′(x)−g′(x), we can substitute x=1 to find C1:
h′(1)=f′(1)−g′(1)
Given that f′(1)=2 and g′(1)=4:
C1=2−4=−2
So, the first derivative equation is:
h′(x)=−2
Step 4: Integrate to find the original function
Now, integrate h′(x)=−2 with respect to x:
h(x)=−2x+C2
(where C2 is another constant of integration)
We know that h(2)=f(2)−g(2). Given f(2)=3 and g(2)=9:
h(2)=3−9=−6
Substitute x=2 into our equation for h(x):
−6=−2(2)+C2
−6=−4+C2
C2=−2
Therefore, the function formula is:
h(x)=−2x−2
Step 5: Evaluate the value at x=4
We need to find the value of f(x)−g(x) at x=4, which is exactly h(4):
h(4)=−2(4)−2
h(4)=−8−2=−10
Correct Answer:
(b) -10
