graph TD
    A[Given: $ax^{2+bx+c}$ > 0 for all x] --> B{Is a > 0?}
    B -->|Yes| C[Need D < 0]
    C --> D[$b^2$ - 4ac < 0]
    D --> E[Solve inequality for parameter k]
    B -->|No or a=0| F[Not possible for all x if a <= 0]
    A2[Given: $ax^{2+bx+c}$ = 0 has real roots] --> G[Need D >= 0]
    G --> H[$b^2$ - 4ac >= 0]
    H --> I[Solve inequality for parameter k]

For "f(x) > 0 for all x": requires a > 0 AND D < 0. This is the most common parametric question in JEE.