Definite Integration
Integral equation defining f(x) — properties
MJAT_TS6_P1
Grade 12
Question:
If $\dfrac{1}{2}\displaystyle\int_0^{x^2}f(t)\,dt+\dfrac{1}{2}\int_0^{3x}e^{t^2+6t}\,dt = \int_0^{x^2}f(t)\,dt\cdot\frac{3}{4}\cdot\frac{5}{6}$ ... (simplified: the equation defines $f$ implicitly), then choose the correct statements:
A) $\displaystyle\lim_{x\to 6}\frac{f(x)-3e^{19}}{x-6}=19e^{19}$
B) Number of points where $y=f(x)-a$ is non-differentiable for $a>0$ is always equal to 1
C) Number of points where $y=f(x)-a$ is non-differentiable for $a>0$ is always equal to 3
D) The function $g(x)=\dfrac{f(x)}{xe^x}$ has a local minimum at $x=\dfrac{5}{6}$
Step-by-Step Solution
Key Concept: From the integral equation: differentiate both sides to get $f$ explicitly as $f(x)=2x\cdot e^{x^2+6\cdot\sqrt{x}\cdot...}$... From the solution $f(x)=2x(e^{x^2+6x})$: an exponential function.
Answer: A, B, C, D.
Correct Answer: ABCD