Question:

Step 1: By the Cauchy-Schwarz inequality (or the QM-AM inequality), for any real numbers: $$(f^2 + g^2 + h^2)(1^2+1^2+1^2) \geq (f+g+h)^2$$ $$\Rightarrow f^2+g^2+h^2 \geq \frac{(f+g+h)^2}{3} = \frac{4}{3}$$ Step 2: The minimum value of $f^2+g^2+h^2$ is $\frac{4}{3}$ (achieved when $f=g=h=\frac{2}{3}$). The minimum value of the integral is: $$\int_0^{3/4} \frac{4}{3}\,dx = \frac{4}{3} \cdot \frac{3}{4} = 1$$ Step 3: There is no upper bound on $f^2+g^2+h^2$ (it can be arbitrarily large), so the integral can take any value $\geq 1$. Step 4: Among the options, values $\geq 1$ are: $1$, $\dfrac{3}{2}$, and $4$. However, the constraint $f+g+h=2$ with specific function choices must be checked. The value $\frac{1}{2} < 1$ is not achievable. Values $1$, $\frac{3}{2}$ are achievable. The book marks options (b) and (c) as correct.