Application of Derivatives & Monotonicity

Differential Geometry Viewpoint

The derivative $f^{\prime }(x_0)$ geometrically represents the instantaneous slope of the tangent line to the curve $y=f(x)$ at $(x_0,f(x_0))$. The second derivative $f^{″}(x_0)$ governs curvature and local convexity.

---

1. Tangents, Normals & Length of Subtangent/Subnormal

📊Visual Mathematical Intuition

Differential Tangent, Normal & Lagrange Mean Value Parallel Chord

Tangent Slope:$m_T = f'(x_0)$

Orthogonal Normal:$m_N = -\frac{1}{f'(x_0)} \implies m_T \cdot m_N = -1$

Lagrange MVT:$f'(c) = \frac{f(b) - f(a)}{b - a} \text{ for some } c \in (a, b)$

Differential tangent slope $f'(x)$, perpendicular normal slope $-1/f'$, sub-tangent/sub-normal segments, and Lagrange Mean Value Theorem chord parallelism $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Let $P(x_1,y_1)$ be a point on $y=f(x)$ with slope $m={\frac{dy}{dx}|}_P$.

• Equation of Tangent: $y-y_1=m(x-x_1)$
• Equation of Normal: $y-y_1=-\frac{1}{m}(x-x_1)$ ($m\ne 0$)
• Length of Tangent: $\left|\frac{y_1\sqrt{1+m^2}}{m}\right|$
• Length of Normal: $|y_1\sqrt{1+m^2}|$
• Length of Subtangent: $\left|\frac{y_1}{m}\right|$
• Length of Subnormal: $|y_{1}m|$

---

2. Rolle's & Lagrange's Mean Value Theorems

Rolle's Theorem

If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists at least one $c\in (a,b)$ such that:

$$f^{\prime }(c)=0$$

Between any two real roots of $f(x)=0$, there lies at least one real root of $f^{\prime }(x)=0$.

Lagrange's Mean Value Theorem (LMVT)

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one $c\in (a,b)$ such that:

$$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$$

---

3. Monotonicity & Extrema

• Strictly Increasing on $I$: $f^{\prime }(x)>0$ for all $x\in I$ (can be $0$ at isolated points).
• Strictly Decreasing on $I$: $f^{\prime }(x)<0$ for all $x\in I$.

First Derivative Test for Local Extrema

• Local Maximum at $x_0$: $f^{\prime }(x)$ changes sign from positive to negative as $x$ increases through $x_0$.
• Local Minimum at $x_0$: $f^{\prime }(x)$ changes sign from negative to positive as $x$ increases through $x_0$.

Second Derivative Test

• If $f^{\prime }(x_0)=0$ and $f^{″}(x_0)<0⟹$ Local Maximum at $x_0$.
• If $f^{\prime }(x_0)=0$ and $f^{″}(x_0)>0⟹$ Local Minimum at $x_0$.

---

4. Concavity & Point of Inflection

• $f^{″}(x)>0⟹$ Concave Upwards (Convex $⌣$, tangent lies below curve).
• $f^{″}(x)<0⟹$ Concave Downwards (Concave $⌢$, tangent lies above curve).
• Point of Inflection: A point where $f^{″}(x)$ changes sign and tangent exists.