Question

Let $f(x)=10x^2+ax,\; x\in \mathbb{R}$ be such that $a^2-400<0$. Let $g(x)=f(x)+f'(x)+f''(x)$. Then $g(x)$ is:

• A. greater than 100 but less than 200
• B. greater than 10 but less than 100
• C. less than 10
• D. greater than 0
• E. less than 1

Hint:

Use discriminant condition to check positivity of quadratic.

Answer 1

The Correct Option is D

Concept:
• Use derivatives and analyze quadratic expression

Step 1: Compute derivatives
\[ f'(x) = 20x + a,\quad f''(x) = 20 \]

Step 2: Form $g(x)$
\[ g(x) = (10x^2 + ax) + (20x + a) + 20 \] \[ = 10x^2 + (a+20)x + (a+20) \]

Step 3: Analyze quadratic
\[ g(x) = 10x^2 + (a+20)x + (a+20) \] Minimum value occurs at vertex.

Step 4: Use condition
\[ a^2 - 400<0 \Rightarrow -20<a<20 \] This ensures quadratic always positive. Final Conclusion:
\[ g(x)>0 \]