Here you will learn second derivative test for maxima and minima with examples. Let’s begin – Second Derivative Test for Maxima and Minima If f(x) is continuous and differentiable at x = a where f'(a) = 0 and f”(a) also exists then for ascertaining maxima/minima at x = a, 2nd dervative can be used (i) …
Here you will learn first derivative test for maxima and minima with examples. Let’s begin – First Derivative Test for Maxima and Minima If f'(x) = 0 at a point (say x = a) and (i) If f'(x) changes sign from positive to negative in the neighbourhood of x = a then x = a …
Here you will learn definitions and concepts of local and absolute maxima and minima class 12. Let’s begin – Maxima and Minima Class 12 Local Maxima A function f(x) is said to have local maxima at x = a if there exist a neighbourhood (a – h, a + h) – {a} such that f(a) …
Here you will learn definition of monotonic function and condition for monotonicity with examples. Let’s begin – Monotonic Function The function f(x) is said to be monotonic on an interval (a, b) if it is either increasing or decreasing on (a, b). A function f(x) is said to be increasing (decreasing) at a point \(x_0\), …
Here you will learn what are increasing and decreasing function with examples. Let’s begin – Increasing and Decreasing Function Strictly Increasing Function A function f(x) is said to be a strictly increasing function on (a, b), if \(x_1\) < \(x_2\) \(\implies\) \(f(x_1)\) < \(f(x_2)\) for all \(x_1\), \(x_2\) \(\in\) (a, b) Thus, f(x) is strictly …
Here you will learn angle of intersection of two curves formula with examples. Let’s begin – Angle of Intersection of Two Curves The angle of intersection of two curves is defined to be the angle between the tangents to the two curves at their point of intersection. Let \(C_1\) and \(C_2\) be two curves having …
Here you will learn equation of tangent and normal to the curve with examples. Let’s begin – Equation of Tangent and Normal to the Curve We know that the equation of line passing through a point \((x_1, y_1)\) and having slope m is \(y – y_1\) = m\((x – x_1)\). and we know that the …
Here you will learn slopes of tangent and normal to the curve with examples. Let’s begin – Slopes of Tangent and Normal to the Curve (a) Slopes of Tangent Let y = f(x) be a continuous curve, and let \(P(x_1, y_1)\) be a point on it. Then, \(({dy\over dx})_P\) is the tangent to the curve …
Here you will learn lagrange’s mean value theorem statement, its geometrical and physical interpretation with examples. Let’s begin – Lagrange’s Mean Value Theorem (LMVT) Statement : Let f be a function that satisfies the following conditions : (i) f is continuous in [a, b] (ii) f is differentiable in (a, b) Then there is a …
Here you will learn mean value theorems i.e rolle’s theorem, lagrange’s theorem and extreme value theorem. Let’s begin – Mean Value Theorems (a) Rolle’s Theorem Let f be a real valued function defined on the closed interval [a, b] such that (i) it is continuous on the closed interval [a, b], (ii) it is differentiable …
