Let $x = -1$ and $x = 2$ be the critical points of the function $f(x) = x^3 + ax^2 + b \log_2|x| + 1,\ x \ne 0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2, -\frac{1}{2}\right]$. Then $|M + m|$ is equal to: (Take $\log_2 2 = 0.7$)

Step 1: Find the derivative of the function

To find the critical points of the function f(x), we first need to compute its derivative, f'(x). The derivative of $$x^3$$ is $$3x^2$$ , the derivative of $$ax^2$$ is $$2ax$$ , and the derivative of $$b \log_2|x|$$ is $$b \cdot \frac{1}{x \ln 2}$$ . The derivative of the constant $$1$$ is $$0$$ .

Step 2: Use critical points to find a and b

We are given that $$x = -1$$ and $$x = 2$$ are critical points. This means that $$f'(x) = 0$$ at these points. Substituting $$x = -1$$ into f'(x) gives equation (1), and substituting $$x = 2$$ gives equation (2).

Step 3: Solve the system of equations for a and b

We have a system of two linear equations with two variables, $$a$$ and $$b$$ . To solve for $$a$$ and $$b$$ , we can multiply equation (1) by 2 to get equation (1'). Then, adding equation (1') and equation (2) eliminates $$a$$ , allowing us to solve for $$b$$ . Once $$b$$ is found, we substitute it back into equation (1) to find $$a$$ .

Step 4: Substitute a and b back into f(x)

Now that we have the values of $$a$$ and $$b$$ , we can substitute them back into the original function f(x). We also use the change of base formula for logarithms, $$\log_2|x| = \frac{\ln|x|}{\ln 2}$$ , to simplify the expression for f(x).

Step 5: Evaluate f(x) at critical points and endpoints

To find the absolute maximum and minimum values in the interval $$\left[-2, -\frac{1}{2}\right]$$ , we need to evaluate f(x) at the endpoints of the interval and at any critical points that lie within the interval. The critical point $$x = -1$$ is in the interval, while $$x = 2$$ is not. We use the given value $$\log_2 2 = 0.7$$ , which implies $$\ln 2 = 0.7 \ln 2 \approx 0.7 \times 0.693 \approx 0.485$$ .

Step 6: Calculate numerical values and find M and m

We calculate the numerical values of f(x) at the points found in the previous step. We use the approximation $$\ln 2 \approx 0.693$$ and the given $$\log_2 2 = 0.7$$ to estimate $$12 \ln 2$$ . Comparing these values, we identify the absolute maximum $$M$$ and absolute minimum $$m$$ in the interval.

Step 7: Calculate |M + m|

Finally, we calculate the absolute value of the sum of the absolute maximum $$M$$ and the absolute minimum $$m$$ .

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