Applications of Derivatives
Equal Roots of Quadratic in m — Finding α+β
nta_pyq_2026_jan
Grade 12
Question:
Let $f:\mathbf{R}\to\mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x)\mathrm{m}^2-2f'(x)\mathrm{m}+f''(x)=0$ in m, has two equal roots for every $x\in\mathbf{R}$. If $f(0)=1$, $f'(0)=2$, and $(\alpha,\beta)$ is the largest interval in which the function $f(\log_e x-x)$ is increasing, then $\alpha+\beta$ is equal to _____
Step-by-Step Solution
Key Concept: Equal roots $\Rightarrow[f'(x)]^2=f(x)f''(x)\Rightarrow f''(x)/f'(x)=f'(x)/f(x)$. Integrating: $f(x)=Ae^{kx}$. Using $f(0)=1,f'(0)=2$: $A=1,k=2$, so $f(x)=e^{2x}$.
$f(x)=e^{2x}$. $g(x)=f(\ln x-x)$ increasing on $(0,1)$. $\alpha+\beta=1$.
Correct Answer: 1