If a and b are constants, such that
`f(x)=asinx+bxcosx+2x^(2)` and f(2)=15, f(-2) is ………. .

Answer

To solve the problem step by step, we start with the function given:

**Step 1: Write down the function and the condition.**
We have:
\[ f(x) = a \sin x + b x \cos x + 2x^2 \]
and we know that:
\[ f(2) = 15 \]

**Step 2: Substitute \( x = 2 \) into the function.**
Now, substituting \( x = 2 \):
\[ f(2) = a \sin(2) + b(2) \cos(2) + 2(2^2) \]
This simplifies to:
\[ f(2) = a \sin(2) + 2b \cos(2) + 8 \]

**Step 3: Set the equation equal to 15.**
Since \( f(2) = 15 \), we can write:
\[ a \sin(2) + 2b \cos(2) + 8 = 15 \]

**Step 4: Rearrange the equation to isolate terms involving \( a \) and \( b \).**
Subtract 8 from both sides:
\[ a \sin(2) + 2b \cos(2) = 15 - 8 \]
This simplifies to:
\[ a \sin(2) + 2b \cos(2) = 7 \] (Equation 1)

**Step 5: Write down the expression for \( f(-2) \).**
Now, we need to find \( f(-2) \):
\[ f(-2) = a \sin(-2) + b(-2) \cos(-2) + 2(-2)^2 \]

**Step 6: Simplify \( f(-2) \) using properties of sine and cosine.**
Using the properties \( \sin(-x) = -\sin(x) \) and \( \cos(-x) = \cos(x) \):
\[ f(-2) = a(-\sin(2)) + b(-2) \cos(2) + 2(4) \]
This simplifies to:
\[ f(-2) = -a \sin(2) - 2b \cos(2) + 8 \]

**Step 7: Substitute the value from Equation 1 into \( f(-2) \).**
From Equation 1, we know:
\[ a \sin(2) + 2b \cos(2) = 7 \]
Thus:
\[ -a \sin(2) - 2b \cos(2) = -7 \]

Now substituting this back into the expression for \( f(-2) \):
\[ f(-2) = -7 + 8 \]

**Step 8: Calculate the final value of \( f(-2) \).**
This simplifies to:
\[ f(-2) = 1 \]

Thus, the final answer is:
\[ f(-2) = 1 \]

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