Question

If \[f(x) = mx + c\], \[f(0) = f'(0) = 1\] , then \[f(2)\]
A) 1
B) 2
C) 3
D) -3

Answer 1

Hint: Here the question is related to the linear equations. On considering the condition for the linear equation we determine the value of m and c. Substituting the value of $m$ and $c$ in the given linear equation, we have a function of x. Then we can determine the value of \[f(2)\].

Complete step by step answer:
A mathematical statement that has an equal to the "=" symbol is called an equation. Linear equations are equations of degree 1.
A linear equation is an equation that is written for two different variables. This equation will be a linear combination of these two variables, and a constant can be present. Surprisingly, when any linear equation is plotted on a graph, it will necessarily produce a straight line - hence the name: Linear equations.
Now consider the given equation \[f(x) = mx + c\]-----(1)
As we have \[f(0) = 1\] ----(2)
On substituting equation (2) in the equation (1) we have
\[ \Rightarrow 1 = m(0) + c\]
On simplifying we have
\[ \Rightarrow c = 1\] -----(3)
Consider \[f'(0) = 1\]---- (4)
Differentiate the equation (1) with respect to x \[f'(x) = m\]----(5)
On substituting the equation (4) in equation (5) we have
\[ \Rightarrow m = 1\] ----(6)
Substitute the equation (3) and the equation (6) to the equation (1)
\[ \Rightarrow f(x) = (1)x + 1\]
On simplifying we have
\[ \Rightarrow f(x) = x + 1\] ---- (7)
Now we determine the value of \[f(2)\]
Substitute the value of x as 2 in the equation (7) we have
\[ \Rightarrow f(2) = 2 + 1\]
On adding we have
\[ \Rightarrow f(2) = 3\]
Hence, option (C) is the correct option.

Note:
In Mathematics, Differentiation can be defined as a derivative of a function with respect to an independent variable. Differentiation, in calculus, can be applied to measure the function per unit change in the independent variable. The derivative formulas which is used in this question is \[\dfrac{d}{{dx}}(x) = 1\] and \[\dfrac{d}{{dx}}(c) = 0\], where c is constant.