Given: [tex]$f(x)=-2 x^3+5 x^2+4 x-3$[/tex]
2.3.1 Solve for [tex]$x$[/tex] if [tex]$f(x)=0$[/tex].
2.3.2 Calculate the coordinates of B and E, the turning points of [tex]$f$[/tex].
Answer (1)
Find the roots of f ( x ) = − 2 x 3 + 5 x 2 + 4 x − 3 = 0 , which are x = − 0.5 , 1 , 3 .
Calculate the first derivative: f ′ ( x ) = − 6 x 2 + 10 x + 4 .
Solve for the critical points by setting f ′ ( x ) = 0 , resulting in x = − 3 1 and x = 2 .
Determine the coordinates of the turning points: ( − 3 1 , − 27 100 ) and ( 2 , 9 ) .
x = − 0.5 , 1 , 3 ; ( − 3 1 , − 27 100 ) , ( 2 , 9 )
Explanation
Problem Analysis We are given the cubic function f ( x ) = − 2 x 3 + 5 x 2 + 4 x − 3 . We need to solve for the roots of f ( x ) = 0 and find the coordinates of the turning points of f ( x ) .
Finding Roots To solve f ( x ) = 0 , we need to find the values of x that satisfy the equation − 2 x 3 + 5 x 2 + 4 x − 3 = 0 . Using a calculator, we find the roots to be x = − 1/2 , x = 1 , x = 3 .
Calculating the First Derivative To find the turning points, we first calculate the first derivative f ′ ( x ) of the function f ( x ) .
f ′ ( x ) = − 6 x 2 + 10 x + 4
Setting the Derivative to Zero Next, we set the first derivative equal to zero, f ′ ( x ) = 0 , and solve for x . This will give the x-coordinates of the turning points. − 6 x 2 + 10 x + 4 = 0
Simplifying the Quadratic Equation We solve the quadratic equation − 6 x 2 + 10 x + 4 = 0 for x . We can simplify this by dividing by -2: 3 x 2 − 5 x − 2 = 0
Factoring the Quadratic Equation We can factor the quadratic equation: ( 3 x + 1 ) ( x − 2 ) = 0
Finding the x-coordinates of Turning Points The solutions for x are: x 1 = − 3 1 , x 2 = 2
Finding the y-coordinates of Turning Points Now, we substitute the x-coordinates of the turning points, x 1 = − 3 1 and x 2 = 2 , back into the original function f ( x ) to find the corresponding y-coordinates. y 1 = f ( − 3 1 ) = − 2 ( − 3 1 ) 3 + 5 ( − 3 1 ) 2 + 4 ( − 3 1 ) − 3 = − 2 ( − 27 1 ) + 5 ( 9 1 ) − 3 4 − 3 = 27 2 + 9 5 − 3 4 − 3 = 27 2 + 15 − 36 − 81 = 27 − 100 y 2 = f ( 2 ) = − 2 ( 2 ) 3 + 5 ( 2 ) 2 + 4 ( 2 ) − 3 = − 2 ( 8 ) + 5 ( 4 ) + 8 − 3 = − 16 + 20 + 8 − 3 = 9
Turning Points Coordinates The coordinates of the turning points are ( − 3 1 , f ( − 3 1 )) = ( − 3 1 , − 27 100 ) and ( 2 , f ( 2 )) = ( 2 , 9 ) .
Final Answer The solutions for x when f ( x ) = 0 are x = − 0.5 , 1 , 3 . The coordinates of the turning points are ( − 3 1 , − 27 100 ) and ( 2 , 9 ) .
Examples
Understanding the roots and turning points of a function is crucial in many fields. For instance, in engineering, you might use this to model the behavior of a system, such as the deflection of a beam under load. The roots would represent points of zero deflection, while the turning points could indicate maximum stress or strain. In economics, these concepts can help model profit curves, where roots are break-even points and turning points indicate maximum profit or minimum loss. By analyzing these key features, engineers and economists can optimize designs and strategies for better performance and efficiency.
