We need to find the maximum value of the function y equals log base 10 of x, plus natural log of x, minus x squared, plus 10. First, let's identify the domain of this function. Since both logarithmic terms require x to be positive, the domain is x greater than zero. Looking at the graph, we can see that the function has a maximum value. To find it precisely, we'll need to use calculus.
To find the maximum value, we need to locate the critical points by finding where the derivative equals zero. The derivative of our function is one over x times natural log of 10, plus one over x, minus two x. Setting this equal to zero gives us our critical point equation. We can simplify this by combining the fractions with a common denominator. Notice on the graph that the derivative, shown in red, crosses the x-axis at the critical point. This is where the original function, shown in blue, reaches its maximum value.
Now, let's solve the equation to find the exact critical point. We rearrange to get the fraction equal to 2x. Multiplying both sides by x times natural log of 10 gives us 1 plus natural log of 10 equals 2x squared times natural log of 10. Solving for x, we get x equals the square root of the fraction 1 plus natural log of 10 over 2 times natural log of 10, which is approximately 0.83. To verify this is a maximum, we check the second derivative, which is negative for all positive x values. Since the second derivative is negative at our critical point, this confirms we have found a maximum value.
Now that we've found the critical point at x equals 0.83, let's calculate the maximum value of our function. Substituting this x-value into our original function, we get log base 10 of 0.83, plus natural log of 0.83, minus 0.83 squared, plus 10. Computing each term, we get negative 0.081 minus 0.186 minus 0.689 plus 10, which equals approximately 9.044. Therefore, the maximum value of our function occurs at x equals 0.83, and the maximum value is approximately 9.044. We can verify this on our graph, where we see the function reaches its peak at this point.
Let's summarize what we've learned. We found the maximum value of the function y equals log base 10 of x plus natural log of x minus x squared plus 10. We identified the critical point by setting the derivative equal to zero and solving the resulting equation. This gave us x equals the square root of the fraction 1 plus natural log of 10 over 2 times natural log of 10, which is approximately 0.83. We confirmed this is a maximum by checking that the second derivative is negative at this point. Finally, we calculated the maximum value of the function to be approximately 9.044. This approach demonstrates how calculus can be used to find extreme values of functions.