Let g(x) = _0^x f(t),dt, where f is a continuous function on [0,3] such that 13 f(t)1 for …

Question

Let $g(x) = \displaystyle\int_0^x f(t)\,dt$, where $f$ is a continuous function on $[0,3]$ such that $\dfrac13\le f(t)\le1$ for all $t\in[0,1]$ and $0\le f(t)\le\dfrac12$ for all $t\in(1,3]$. The largest possible interval in which $g(3)$ lies is:

Accepted Answer

$$g(3) = \int_0^1 f(t)\,dt + \int_1^3 f(t)\,dt$$

Bounding each piece:

$$\int_0^1\frac13\,dt + \int_1^3 0\,dt \le g(3) \le \int_0^1 1\,dt + \int_1^3 \frac12\,dt$$

$$\frac13 \le g(3) \le 1+1 = 2$$

So $g(3) \in \left[\dfrac13, 2\right]$.

Solutions · 1