Question

How do you find Maximum In Calculus?

Answer 1

We can find the maximum, minimum values of functions by two methods:

1. By plotting the graph of the function
2. By using derivatives of the function.

Method 1. A maximum is a high point and a minimum is a low point of the graph. In a smoothly changing function a maximum or minimum is always where the value of slope is zero (except for a saddle point) and the value of slope can be found by the derivative of the function.

Saddle point: A saddle point is a point at which the function has neither a maximum nor a minimum value.

Method 2. Take $f\left(x\right)$ to be a function of $x$. Then the value of $x$ for which the derivative of $f\left(x\right)$ with respect to $x$ is equal to zero corresponds to a maximum, a minimum or an inflexion point of the function $f\left(x\right)$.

The second derivative demonstrates whether a point with zero first derivative is maximum, a minimum, or an inflexion point,

For a maximum, $\frac{dy}{dt}=0$and $\frac{d^{2}y}{d{x}^2}<0$ that means the slope of the curve is at first positive, then goes through zero to become negative.

For a minimum, $\frac{dy}{dt}=0$and $\frac{d^{2}y}{d{x}^2}>0$ that means the slope of the curve is at first negative, then goes through zero to become positive.

For an inflexion point, $\frac{dy}{dt}=0$and $\frac{d^{2}y}{d{x}^2}=0$ It represents a point where the curvature is changing its sense.

Hence, we can find the maximum by derivative or by plotting graph of the given function.