20 19 The comprehensive application part of the national mathematics volume of college entrance examination is a difficult topic, which requires candidates to have high comprehensive application ability of mathematics. Let's make a detailed analysis on this topic.

Title description

The function $ f (x) = \ frac {1} {2} x 4-2x2+ax+b $ is known, where $ a and b $ are constants, and the tangent equation of $f(x)$ at $x=- 1$ is

Solution thinking

This topic requires us to deduce the function and solve the extreme value of the function. The specific steps are as follows:

1. Take the derivative of $f(x)$ and get $ f' (x) = 2x 3-4x $.

2. Substitute $x=- 1$ into $f'(x)$ to get $f'(- 1)=-6$.

3. Because the tangent equation of $f(x)$ at $x=- 1$ is $y=3x-4$, $f'(- 1)=3$.

4. Because the derivative of $f'(x)$ is $-6$ at $x=- 1$, $x=- 1$ is a maximum point of $f(x)$.

5. Because the derivative of $f'(x)$ is $0$ when $x=0$, $x=0$ is the minimum point of $f(x)$.

6. Substitute $x=0$ into $f(x)$ to get $f(0)=b$.

7. Therefore, the minimum value of $f(x)$ is $b$.

Answer analysis

According to the above steps, we can conclude that the minimum value of $f(x)$ is $b$. Because the specific value of $b$ is not given in the title, it is impossible to get the answer directly. However, we can solve the value of $b$ in other ways.

Since the tangent equation of $f(x)$ at $x=- 1$ is $y=3x-4$, $ f (-1) = \ frac {1} {2} (-1) Substituting $b$, we get $b=3a-3$.

Substitute $b=3a-3$ into $f(x)$ to get $ f (x) = \ frac {1} {2} x 4-2x2+ax+3a-3 $. Substitute $x=0$ into $f(x)$ to get $f(0)=3a-3$. Because the minimum value of $f(x)$ is $f(0)$, the minimum value of $f(x)$ is $3a-3$.

To sum up, the minimum value of $f(x)$ is $3a-3$. Because the specific value of $a$ is not given in the title, it is impossible to get the answer directly. However, we can solve the value of $a$ in other ways.