Application of Derivatives — JEE Mains Mathematics

Formulas (2)

• Rates of change · Chain rule for rates
  $$\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}$$
• Tangents and normals · Tangent at a point
  $$y-y_0=f'(x_0)(x-x_0)$$

Watch out for (2)

• Differentiate before substituting
  → Rates of change
• A normal parallel to a line
  → Tangents and normals

Formulas (3)

• The sign of the derivative · Monotonicity test
  $$f'(x)>0\ \text{on }I\ \Rightarrow\ f\ \text{increasing on }I$$
• Parameters that make a function monotonic · Quadratic never negative
  $$ax^2+bx+c\ge0\ \forall x\ \Leftarrow\ a>0,\ b^2-4ac\le0$$
• Increasing derivatives and order · Symmetric sum
  $$g(x)=f(x)+f(c-x):\ g'(x)=f'(x)-f'(c-x)$$

Watch out for (3)

• Each piece is not the union
  → The sign of the derivative
• Include the boundary value
  → Parameters that make a function monotonic
• f″ > 0 is about f′, not f
  → Increasing derivatives and order

Formulas (2)

• A monotonic function has at most one root · At most one root
  $$f'(x)>0\ \forall x\ \Rightarrow\ f(x)=0\ \text{has at most one root}$$
• Counting roots from the extreme values · Three roots of a cubic
  $$f(x)=k\ \text{has 3 distinct roots}\ \Leftarrow\ f_{\min}<k<f_{\max}$$

Watch out for (2)

• At most one is not exactly one
  → A monotonic function has at most one root
• Count the ends too
  → Counting roots from the extreme values

Formulas (2)

• Sign changes of the derivative · Derivative of an integral
  $$\frac{d}{dx}\int_a^{g(x)}h(t)\,dt=h\big(g(x)\big)\,g'(x)$$
• Corners, cusps and piecewise functions · Critical points
  $$f'(c)=0\ \text{or}\ f'(c)\ \text{does not exist}\ (c\in\text{domain})$$

Watch out for (2)

• Even powers do not change sign
  → Sign changes of the derivative
• A corner need not be an extremum
  → Corners, cusps and piecewise functions

Formulas (2)

• A polynomial from its extrema · Start from the derivative
  $$f'(x)=k(x-a)(x-b)\ \Rightarrow\ f(x)=k\left(\tfrac{x^3}3-\tfrac{a+b}2x^2+abx\right)+C$$
• Where a cubic's maximum and minimum sit · Order of the extrema
  $$f(x)=ax^3+\dots,\ a>0:\ x_{\max}<x_{\min}$$

Watch out for (2)

• Use every condition once
  → A polynomial from its extrema
• Check the sign of the leading term
  → Where a cubic's maximum and minimum sit

Formulas (2)

• Closed interval: critical points and endpoints · Closed-interval method
  $$\max_{[a,b]}f=\max\{f(a),\,f(b),\,f(c):f'(c)=0\}$$
• Ranges and least values over a whole domain · AM-GM
  $$a+b\ge2\sqrt{ab}\quad(a,b>0),\ \text{equality when }a=b$$

Watch out for (2)

• The endpoints count
  → Closed interval: critical points and endpoints
• Is the bound reached?
  → Ranges and least values over a whole domain

Formulas (2)

• Largest area or volume · The method
  $$Q=Q(x),\quad Q'(x)=0,\quad Q''(x)<0\ \Rightarrow\ \text{maximum}$$
• Shortest distances · Nearest point to a line
  $$f'(x_0)=\text{slope of the line}$$

Watch out for (2)

• Stay inside the allowed range
  → Largest area or volume
• Only if they do not meet
  → Shortest distances

Formulas (2)

• Rolle's theorem and the mean value theorem · Mean value theorem
  $$f'(c)=\frac{f(b)-f(a)}{b-a},\quad c\in(a,b)$$
• Zeros of f′ and f″ · Rolle, repeated
  $$n\ \text{zeros of }f\ \Rightarrow\ \ge n-1\ \text{zeros of }f'$$

Watch out for (2)

• Check the hypotheses
  → Rolle's theorem and the mean value theorem
• The zeros must be distinct
  → Zeros of f′ and f″