Exercise · ZW-FUNDAMENTAL-THEOREM-OF-CALCULUS-017 · ★★☆ Intermediate
Let \(F(x)=\displaystyle\int_0^x (t^2-4)\,dt\). At which \(x>0\) does \(F\) have a local minimum?
Choose one option.
Show answer
- A. \(x=2\)
Explanation
Since \(F'(x)=x^2-4\), the critical points are \(\pm2\). At \(x=2\), \(F'\) goes from negative to positive: a local minimum. At \(x=-2\) there is a maximum.
Review the concept: Fundamental theorem of calculus
It links derivatives and integrals: a definite integral equals an antiderivative evaluated at the endpoints, and differentiating an integral with variable upper limit returns the integrand.
\[\begin{gathered}\int_a^b f(x)\,dx=F(b)-F(a)\\ \frac{d}{dx}\int_a^x f(t)\,dt=f(x)\end{gathered}\]
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