Applications of Derivatives Question - None

Question

Paragraph for Question nos. 668 to 669
Let $\alpha < \beta < \gamma$ be three numbers in G.P. Let $f(x) = x^3 - ax^2 + bx - 8$ be a polynomial such that $f(x) = 0$ has three roots $\alpha$, $\beta$ and $\gamma$, where $\alpha$, $\gamma$ are integers.

Let $y = g(x)$ be a twice differentiable function such that $g(\alpha) = 0$, $g(\beta) = 2$, $g(\gamma) = -3$, $g(a) = 5$, $g(b) = 0$, then number of minimum distinct real roots of the equation $(g'(x))^2 + g(x) \cdot g''(x) = 0$ in $[1, 14]$ is:

Accepted Answer

Step 1: Rewrite the equation: (g'(x))² + g(x)·g''(x) = 0 can be recognized as d/dx[g(x)·g'(x)] = 0, since d/dx[g(x)·g'(x)] = (g'(x))² + g(x)·g''(x).

Step 2: This means g(x)·g'(x) has critical points, so either g'(x) = 0 or g(x) = 0.

Step 3: Identify zeros of g(x) in [1,14]: g(α) = 0, g(β) = 2 (not zero), g(γ) = -3 (not zero), g(a) = 5 (not zero), g(b) = 0. So g has at least zeros at α and b.

Step 4: Apply Rolle's theorem between consecutive zeros and extrema:
• Between α and b where g(α) = 0 and g(b) = 0: g'(x) must equal 0 at least once (1 root)
• Between α and the point where g achieves maximum/minimum (from the given values): at least one more critical point
• Between regions where g changes sign: additional roots guaranteed

Step 5: From the given data points with values 0, 2, -3, 5, 0, Rolle's theorem applied between consecutive extrema and zeros guarantees at least 4 distinct real roots of g'(x) = 0, hence 4 distinct roots of the original equation.

∴ Answer: D

Question:

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