a) Write the relation between marginal revenue and total revenue. [1] b) Define stationary point for a function. [1] c) What is monotonic function ? Write it. [1] d) For a function f(x) , f''(x) 0…

a) Relation between Marginal Revenue (MR) and Total Revenue (TR)

Marginal Revenue is defined as the derivative of Total Revenue with respect to quantity sold. Mathematically, it is expressed as:

$$\text{MR}=\frac{d(\text{TR})}{dQ}$$

Alternatively, it can also be written as the difference in Total Revenue when the quantity changes by one unit:

$$\text{MR}={\text{TR}}_{Q+1}-{\text{TR}}_Q$$

b) Stationary Point for a Function

A stationary point (or critical point) of a function $f(x)$ is a point in its domain where the first derivative $f^{\prime }(x)$ is either zero or undefined. At such points, the function may have a local maximum, local minimum, or a saddle point.

Mathematically: $$f^{\prime }(x)=0\text{or}{f}^{\prime }(x)\text{ is undefined}$$

c) Monotonic Function

A monotonic function is a function that preserves the order of its inputs. It is either entirely non-increasing or non-decreasing over its domain.

• Monotonically Increasing: If $x_1<x_2$, then $f(x_1)\le f(x_2)$.
• Monotonically Decreasing: If $x_1<x_2$, then $f(x_1)\ge f(x_2)$.

Example:

• $f(x)=x^3$ is monotonically increasing.
• $f(x)=-x^2$ is monotonically decreasing on $x\le 0$.

d) Nature of $f(x)$ when $f^{\prime \prime }(x)>0$ and $f^{\prime }(x)=0$

Given:

• $f^{\prime \prime }(x)>0$ (second derivative is positive)
• $f^{\prime }(x)=0$ (first derivative is zero)

This implies that the function $f(x)$ has a local minimum at that point.

e) Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating:

1. First Part: If $f$ is continuous on $[a,b]$ and $F$ is defined by: $$F(x)={\int }_a^{x}f(t)dt$$ then $F$ is differentiable on $(a,b)$ and: $$F^{\prime }(x)=f(x)$$

2. Second Part: If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$ on $[a,b]$, then: $${\int }_a^{b}f(x)dx=F(b)-F(a)$$