IPMAT Indore 2025 · Quantitative Ability · Question 29

IPMAT Indore 2025 previous year question on Maxima and Minima, with the answer and a step-by-step solution.

Maxima and MinimaAlgebraMultiple choiceEasy
Let f\left( x \right)={a}^{2}{x}^{2}+2bx+c where, a\ne 0, b, c are real numbers and x is a real variable then
  1. A
    f(x) has a maximum and a minimum
  2. B
    f(x) has a minimum and no maximum
  3. C
    f(x) has a maximum and no minimum
  4. D
    f(x) has no minimum and no maximum
Pick an option, then check.
Answer & solution

Answer: Option B

Step 1 — look at the leading coefficient. The coefficient of {x}^{2} is {a}^{2}. Since a\ne 0 and a is real,

{a}^{2}>0 \text{always}

This is the whole question — the square guarantees a positive leading coefficient whatever a is.

Step 2 — what a positive leading coefficient means. The graph is a parabola opening upwards. Such a parabola:

has a lowest point at its vertex → a minimum exists

rises without bound as x\to \pm \infty → no maximum

Step 3 — the minimum value, for completeness, occurs at x=-\frac{b}{{a}^{2}} and equals c-\frac{{b}^{2}}{{a}^{2}}.

Answer: Option 2 — f\left( x \right) has a minimum and no maximum

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