Let x=−1 and x=2 be the critical points of the function $f(x) = x^3 + ax^2 + b \log_2|x| + 1,\ x
e 0.LetmandMrespectivelybetheabsoluteminimumandtheabsolutemaximumvaluesoffintheinterval\left[-2, -\frac{1}{2}\right].Then∣M+m∣isequalto:(Take\log_2 2 = 0.7$)
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Step-by-Step Solution
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
x3
is
3x2
, the derivative of
ax2
is
2ax
, and the derivative of
blog2∣x∣
is
b⋅xln21
. 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,
log2∣x∣=ln2ln∣x∣
, 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
[−2,−21]
, 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
log22=0.7
, which implies
ln2=0.7ln2≈0.7×0.693≈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
ln2≈0.693
and the given
log22=0.7
to estimate
12ln2
. 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
.