Maxima-minima Formula Sheet

Maxima and minima of a function represent the highest and lowest points, respectively, on its graph, indicating where the function reaches its peak or bottommost value within a given interval.

Neetesh Kumar | June 02, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Introduction
• 2. Derivative test for ascertaining maxima/minima
• 3. Useful formulae of mensuration to remember
• 4. Significance of the sign of 2nd order derivative
• 5. Some special points on a curve
• 6. Some standard results
• 7. Minimum and Maximum distance between two Curves

1. Introduction:

Maxima & Minima:

(a) Local Maxima or Relative Maxima:
A function $f(x)$ is said to have a local maxima at $x = a$ if $f(a) > f(x) \, \forall \, x \in (a - h, a + h) \cap D_f$
Where $h$ is some positive real number.

(b) Local Minima or Relative Minima: A function $f(x)$ is said to have a local minima at $x = a$ if $f(a) < f(x) \, \forall \, x \in (a - h, a + h) \cap D_f$
Where $h$ is some positive real number.

(c) Absolute Maxima (Global Maxima):
A function $f$ has absolute maxima (or global maxima) at $c$ if $f(c) \geq f(x)$ for all $x$ in $D$, where $D$ is the domain of $f$. The number $f(c)$ is the maximum value of $f$ on $D$.

(d) Absolute Minima (Global Minima): A function $f$ has an absolute minimum at $c$ if $f(c) \leq f(x)$ for all $x$ in $D$ and the number $f(c)$ is called the minimum value of $f$ on $D$.

Note:

1. The term 'extrema' is used for both maxima and minima.
2. A function's local maximum (minimum) value may not be the greatest (least) value in a finite interval.
3. A function can have several extreme values such that a local minimum value may be greater than a local maximum value.
4. It is unnecessary that $f(x)$ always has local maxima/minima at the endpoints of the given interval when they are included.
   
   

2. Derivative test for ascertaining maxima/minima:

(a) First Derivative Test:

If $f'(x) = 0$ at a point (say $x = a$) and:

• If $f'(x)$ changes sign from positive to negative in the neighbourhood of $x = a$ then $x = a$ is said to be a point of local maxima.
• If $f'(x)$ changes sign from negative to positive in the neighbourhood of $x = a$ then $x = a$ is said to be a point of local minima.

Note: If $f'(x)$ does not change sign, i.e., has the same sign in a certain complete neighborhood of $a$, then $f(x)$ is either increasing or decreasing throughout this neighborhood, implying that $x = a$ is not a point of extremum of $f$.

(b) Second Derivative Test:

If $f(x)$ is continuous and differentiable at $x = a$ where $f'(a) = 0$ (stationary points) and $f''(a)$ also exists, then for ascertaining maxima/minima at $x = a$, the second derivative test can be used:

• If $f''(a) > 0$ then $x = a$ is a point of local minima.
• If $f''(a) < 0$ then $x = a$ is a point of local maxima.

(c) nth Derivative Test:

Let $f(x)$ be a function such that $f'(a) = f''(a) = f'''(a) = \ldots = f^{(n-1)}(a) = 0$ & $f^{(n)}(a) \neq 0$, then:

• If $n$ is even:
  • $f^{(n)}(a) > 0 \implies$ Minima
  • $f^{(n)}(a) < 0 \implies$ Maxima
• If $n$ is odd then neither maxima nor minima at $x = a$.

3. Useful formulae of mensuration to remember:

• Volume of a cuboid = $lbh$.
• Surface area of a cuboid = $2(lb + bh + hl)$.
• Volume of a prism = area of the base $\times$ height.
• Lateral surface area of a prism = perimeter of the base $\times$ height.
• Total surface area of a prism = lateral surface area $+$ 2 $\times$ area of the base.
• Volume of a pyramid = $\frac{1}{3}$ area of the base $\times$ height.
• Curved surface area of a pyramid = $\frac{1}{2}$ (perimeter of the base) $\times$ slant height.
• Volume of a cone = $\frac{1}{3}\pi r^2 h$.
• Curved surface area of a cylinder = $2 \pi r h$.
• Total surface area of a cylinder = $2 \pi r h + 2 \pi r^2$.
• Volume of a sphere = $\frac{4}{3} \pi r^3$.
• Surface area of a sphere = $4 \pi r^2$.
• Area of a circular sector = $\frac{1}{2}r^2 \theta$, when $\theta$ is in radians.
• Perimeter of a circular sector = $2r + r\theta$.
  
  

4. Significance of the sign of 2nd order derivative:

The sign of the 2nd order derivative determines the concavity of the curve.
If $f''(x) > 0 \, \forall \, x \in (a, b)$ then the graph of $f(x)$ is concave upward in $(a, b)$.
Similarly, if $f''(x) < 0 \, \forall \, x \in (a, b)$ then the graph of $f(x)$ is concave downward in $(a, b)$.

5. Some special points on a curve:

• Stationary Points:
  The stationary points are the domain points where $f'(x) = 0$.

• Critical Points:
  There are three kinds of critical points:

  • The point at which $f'(x) = 0$
  • The point at which $f'(x)$ does not exist
  • The endpoints of the interval (if included)

  These points belong to the domain of the function.

  Note: Local maxima and local minima occur at critical points only, but not all critical points will correspond to local maxima/local minima.

• Point of Inflection:
  A point of inflection is a point where the graph of a function has a tangent line and where the strict concavity changes.
  For finding the point of inflection of any function, compute the points (x-coordinate) where $\frac{d^2y}{dx^2} = 0$ or $\frac{d^2y}{dx^2}$ does not exist.
  Let the solution be $x = a$, if $\frac{d^2y}{dx^2} = 0$ at $x = a$ and the sign of $\frac{d^2y}{dx^2}$ changes about this point, then it is called a point of inflection.
  
  

6. Some standard results:

• The rectangle of the largest area inscribed in a circle is a square.

• The function $y = \sin(mx) \cos(nx)$ attains the maximum value at $x = -\frac{1}{m}\tan^{-1}(\frac{n}{m})$

• If $0 < a < b$ then $|x - a| + |x - b| \geq b - a$ and equality holds when $x \in [a, b]$.

• If $0 < a < b < c$ then $|x - a| + |x - b| + |x - c| \geq c - a$ and equality holds when $x = b$.

• If $0 < a < b < c < d$ then $|x - a| + |x - b| + |x - c| + |x - d| \geq d - a$ and equality holds when $x \in [b, c]$.
  
  

7. Minimum and Maximum distance between two Curves:

The least/greatest distance between two non-intersecting curves usually lies along the common normal (wherever defined).

Note:
Given a fixed point $A(a, b)$ and a moving point $P(x, f(x))$ on the curve $y = f(x)$. Then $AP$ will be the maximum or minimum if it is normal to the curve at $P$.

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