1. **Problem Statement:** We analyze the function $f$ defined on the interval $[1,5]$ with the given curve characteristics. 2. **Given Information:** - The curve has two local maxima and one local minimum between them. - The function starts near the x-axis at $x=1$ and returns near the x-axis at $x=5$. 3. **Key Concepts:** - **Critical points** occur where $f'(x)=0$ or $f'(x)$ is undefined; these correspond to local maxima, minima, or saddle points. - **Inflection points** occur where the concavity changes, i.e., where $f''(x)=0$ and the sign of $f''(x)$ changes. - **Absolute maximum** is the highest value of $f$ on the interval. - **Convex upward** means $f''(x)>0$ on the interval. 4. **Analyzing the curve:** - The curve has two peaks (local maxima) and one valley (local minimum), so there are three critical points. - The concavity changes twice (from convex to concave or vice versa) indicating two inflection points. - The highest peak is an absolute maximum value on $[1,5]$. - The curve is not convex upward on the entire open interval $(1,5)$ because it has both concave upward and concave downward parts. 5. **Conclusion:** - (a) True: three critical points. - (b) True: two inflection points. - (c) True: absolute maximum exists. - (d) False: the curve is not convex upward on the entire interval $(1,5)$. **Final answer:** The statement (d) is not correct.