Function: Increasing and Decreasing

Increasing Function

• Y increases as X increases.
• Mathematically: If X1 < X2 then F(X1) ≤ F(X2).
• A function is considered increasing if the first derivative is positive.

Decreasing Function

• Y decreases as X increases.
• Mathematically: If X1 < X2 then F(X1) ≥ F(X2).
• A function is considered decreasing if the first derivative is negative.

Question-Solving Application

• By finding the first derivative, we determine whether the function is increasing or decreasing.
• For example, F(x) = sin^2(x) - 2cos(x): Check by finding the first derivative.

Tangent and Normal

• Slope of Tangent: dy/dx = first derivative at the point.
• Slope of Normal: -1/(dy/dx)
• Equation of Tangent: y - y_0 = m(x - x_0)
• Equation of Normal: y - y_0 = -1/m(x - x_0)

Application of Tangent and Normal

• To find the slope of the tangent and normal line at a point, use the first derivative and its negative reciprocal.

Maxima and Minima

First Derivative Test

1. Find the first derivative of the function F(x).
2. Set the first derivative equal to zero and find the critical points (c_1, c_2).
3. Check the sign change:
   • If the first derivative changes from negative to positive at a point, it is a minima.
   • If the first derivative changes from positive to negative at a point, it is a maxima.

Second Derivative Test

1. Find F''(c):
   • If F''(c) > 0, it is a minima.
   • If F''(c) < 0, it is a maxima.

Question Solving

• Example: Determine maxima or minima using the first and second derivatives for F(x) = x^2 + 3x - 4.

Point of Inflection

• A point where the concavity of the function changes.