If

𝑓
(
𝑥
)
=
∫
0
𝑥
(
7
𝑡
3
+
5
𝑡
2
−
8
𝑡
)
d
𝑡
,

then which of the following is
𝑓
″
(
𝑥
)
?

Question

If

𝑓
(
𝑥
)
=
∫
0
𝑥
(
7
𝑡
3
+
5
𝑡
2
−
8
𝑡
)
d
𝑡
,

then which of the following is 
𝑓
″
(
𝑥
)
?

BlackTom AI · Accepted Answer

The Fundamental Theorem of Calculus states that if f(x) = ∫_0^x g(t) dt, then f'(x) = g(x). Differentiation again uses ordinary power-rule differentiation.

1) Relevant knowledge points and concepts
- Fundamental Theorem of Calculus (FTC): d/dx ∫_a^x g(t) dt = g(x).
- Power rule for derivatives: d/dx [c x^n] = c n x^(n-1).
- Linearity of differentiation.

2) Complete reasoning process step by step
Given f(x) = ∫_0^x (7 t^3 + 5 t^2 − 8 t) dt.
Apply FTC to get f'(x) by evaluating the integrand at t = x:
f'(x) = 7 x^3 + 5 x^2 − 8 x.

Differentiate f'(x) to obtain f''(x) using the power rule term-by-term:
- d/dx [7 x^3] = 7 * 3 x^2 = 21 x^2
- d/dx [5 x^2] = 5 * 2 x = 10 x
- d/dx [−8 x] = −8

3) Intermediate calculation steps (alternative verification by integrating first)
Integrate the integrand to find f(x) explicitly:
∫(7 t^3) dt = 7 * (t^4/4) = (7/4) t^4
∫(5 t^2) dt = 5 * (t^3/3) = (5/3) t^3
∫(−8 t) dt = −8 * (t^2/2) = −4 t^2
So f(x) = (7/4) x^4 + (5/3) x^3 − 4 x^2 (lower limit 0 contributes 0).
Differentiate twice:
f'(x) = 7 x^3 + 5 x^2 − 8 x (matches FTC result),
f''(x) = 21 x^2 + 10 x − 8.

4) Why the correct answer is correct
Applying FTC gives f'(x) exactly as the integrand evaluated at x, and the second derivative is obtained by standard differentiation of that polynomial. Coefficients result directly from multiplying exponents by original coefficients.

5) Why other options are incorrect
No other answer choices are provided here; common incorrect alternatives would come from arithmetic mistakes (wrong coefficients like 14x^2 or missing the −8), or from differentiating the integral incorrectly (e.g., treating the integral as constant).

6) Conclusion
Thus f''(x) = 21 x^2 + 10 x − 8, which matches the provided correct answer.

If 𝑓 ( 𝑥 ) = ∫ 0 𝑥 ( 7 𝑡 3 + 5 𝑡 2 − 8 𝑡 ) d 𝑡 , then which of the following is 𝑓 ″ ( 𝑥 ) ?单项选择题

类似问题

If 𝑓 ( 9 ) = 6 , 𝑓 ′ is continuous, and ∫ 3 9 𝑓 ′ ( 𝑡 ) 𝑑 𝑡 = 4 , what is the value of 𝑓 ( 3 ) ?

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If 𝑓 ( 𝑥 ) = ∫ 0 𝑥 ( 3 𝑡 3 + 7 𝑡 2 + 2 𝑡 ) d 𝑡 , then which of the following is 𝑓 ″ ( 𝑥 ) ?

If 𝑓 ( 16 ) = 14 , 𝑓 ′ is continuous, and ∫ 0 16 𝑓 ′ ( 𝑡 ) 𝑑 𝑡 = 17 , what is the value of 𝑓 ( 0 ) ?

Find the derivative.

Let [math: F(x)=∫03xet2 dt]\displaystyle F(x)=\int _0^{3x}{e^{t^2} dt}. Compute [math: F′(0.5)]F’(0.5). (Correct the answer to 2 decimal places.)

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