How to Find Stationary Points for the Function ( f(x) 2x^2 - 5x - 3 )

In calculus, finding the stationary points of a function is an important concept used in optimization problems. In this guide, we will walk through the process of finding the stationary points of the function ( f(x) 2x^2 - 5x - 3 ), using a step-by-step approach.

Step 1: Expand the Function

First, expand the given function:

f(x) 2x2- 5x- 3

Step 2: Differentiate the Function

To find the stationary points, we need to differentiate the function with respect to ( x ). Using the power rule, we get:

f′(x) 4x-5

Step 3: Set the Derivative Equal to Zero

Set the derivative equal to zero to find the stationary points:

4x-50

Solving for ( x ):

x 5 4

Step 4: Solve for the ( y )-Coordinate

To find the corresponding ( y )-coordinate, substitute ( x frac{5}{4} ) back into the original function:

f( 5 4 ) ( 5 4 - 3 )( 2#x00D7; 5 4 - 1 )

Simplifying:

f( 5 4 ) ( 5 4 - 12 4 )( 10 4 - 4 4 ) ( -7 4 )( 14 4 ) - 98 16 - 49 8

Thus, the stationary point is at ( left(frac{5}{4}, -frac{49}{8}right) ).

Determine the Nature of the Stationary Point

To determine if the stationary point is a minimum or maximum, we can use the second derivative test:

f#x2032;#x2032;(x)4

Since the second derivative ( f''(x) 4 ) is positive, the stationary point at ( x frac{5}{4} ) is a minimum point.

Conclusion

We have successfully found that the stationary point for the function ( f(x) 2x^2 - 5x - 3 ) is at ( left(frac{5}{4}, -frac{49}{8}right) ). This point is a minimum, as confirmed by the second derivative test.