Final answer:
To find the length of the graph of f(x)=14x²-12ln(x) for 1≤x≤9, integrate the square root of 1 plus the square of the derivative of the function.
Step-by-step explanation:
The length of the graph of f(x)=14x²-12ln(x) for 1≤x≤9 can be found by using integration. To find the length, we need to integrate the square root of 1 plus the square of the derivative of the function.
The derivative of f(x)=14x²-12ln(x) is f'(x)=28x-12/x. Taking the derivative of f'(x) and squaring it, we get (f''(x))²=(28+12/x²)².
Integrating the square root of 1+(f''(x))² from 1 to 9 will give us the length of the graph. The integral can be evaluated using numerical methods or software.