Answer :
To solve the problem, we need to find out what [tex]\( f''(3) \)[/tex] is, given that [tex]\( f'(x) = x^2 f(x) \)[/tex] and [tex]\( f(3) = 2 \)[/tex].
### Step-by-Step Solution
1. Differentiate [tex]\( f'(x) = x^2 f(x) \)[/tex]:
We need to find the second derivative of [tex]\( f(x) \)[/tex]. Starting with the first derivative given:
[tex]\[
f'(x) = x^2 f(x)
\][/tex]
Differentiate both sides with respect to [tex]\( x \)[/tex]. The right-hand side requires the product rule, as it is a product of [tex]\( x^2 \)[/tex] and [tex]\( f(x) \)[/tex]. The product rule states that [tex]\( (uv)' = u'v + uv' \)[/tex].
Let [tex]\( u = x^2 \)[/tex] and [tex]\( v = f(x) \)[/tex]. Then, [tex]\( u' = 2x \)[/tex] and [tex]\( v' = f'(x) \)[/tex].
Applying the product rule:
[tex]\[
f''(x) = \frac{d}{dx}(x^2 f(x)) = u'v + uv' = 2x f(x) + x^2 f'(x)
\][/tex]
2. Substitute [tex]\( f(3) = 2 \)[/tex] and the expression for [tex]\( f'(3) \)[/tex]:
Given [tex]\( f(3) = 2 \)[/tex], this value is substituted for [tex]\( f(x) \)[/tex] when [tex]\( x = 3 \)[/tex].
We also know that:
[tex]\[
f'(x) = x^2 f(x)
\][/tex]
So,
[tex]\[
f'(3) = 3^2 \cdot f(3) = 9 \cdot 2 = 18
\][/tex]
We need [tex]\( f''(3) \)[/tex]:
[tex]\[
f''(x) = 2x f(x) + x^2 f'(x)
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] and the values [tex]\( f(3) = 2 \)[/tex] and [tex]\( f'(3) = 18 \)[/tex]:
[tex]\[
f''(3) = 2 \cdot 3 \cdot 2 + 3^2 \cdot 18
\][/tex]
3. Calculate [tex]\( f''(3) \)[/tex]:
[tex]\[
f''(3) = 2 \cdot 3 \cdot 2 + 9 \cdot 18
\][/tex]
Simplify the calculations:
[tex]\[
f''(3) = 12 + 162
\][/tex]
Thus,
[tex]\[
f''(3) = 174
\][/tex]
4. Conclusion:
Therefore, the value of [tex]\( f''(3) \)[/tex] is [tex]\(\boxed{174}\)[/tex].
### Step-by-Step Solution
1. Differentiate [tex]\( f'(x) = x^2 f(x) \)[/tex]:
We need to find the second derivative of [tex]\( f(x) \)[/tex]. Starting with the first derivative given:
[tex]\[
f'(x) = x^2 f(x)
\][/tex]
Differentiate both sides with respect to [tex]\( x \)[/tex]. The right-hand side requires the product rule, as it is a product of [tex]\( x^2 \)[/tex] and [tex]\( f(x) \)[/tex]. The product rule states that [tex]\( (uv)' = u'v + uv' \)[/tex].
Let [tex]\( u = x^2 \)[/tex] and [tex]\( v = f(x) \)[/tex]. Then, [tex]\( u' = 2x \)[/tex] and [tex]\( v' = f'(x) \)[/tex].
Applying the product rule:
[tex]\[
f''(x) = \frac{d}{dx}(x^2 f(x)) = u'v + uv' = 2x f(x) + x^2 f'(x)
\][/tex]
2. Substitute [tex]\( f(3) = 2 \)[/tex] and the expression for [tex]\( f'(3) \)[/tex]:
Given [tex]\( f(3) = 2 \)[/tex], this value is substituted for [tex]\( f(x) \)[/tex] when [tex]\( x = 3 \)[/tex].
We also know that:
[tex]\[
f'(x) = x^2 f(x)
\][/tex]
So,
[tex]\[
f'(3) = 3^2 \cdot f(3) = 9 \cdot 2 = 18
\][/tex]
We need [tex]\( f''(3) \)[/tex]:
[tex]\[
f''(x) = 2x f(x) + x^2 f'(x)
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] and the values [tex]\( f(3) = 2 \)[/tex] and [tex]\( f'(3) = 18 \)[/tex]:
[tex]\[
f''(3) = 2 \cdot 3 \cdot 2 + 3^2 \cdot 18
\][/tex]
3. Calculate [tex]\( f''(3) \)[/tex]:
[tex]\[
f''(3) = 2 \cdot 3 \cdot 2 + 9 \cdot 18
\][/tex]
Simplify the calculations:
[tex]\[
f''(3) = 12 + 162
\][/tex]
Thus,
[tex]\[
f''(3) = 174
\][/tex]
4. Conclusion:
Therefore, the value of [tex]\( f''(3) \)[/tex] is [tex]\(\boxed{174}\)[/tex].
