Applications of Derivatives
Decreasing interval of g(x) from f(x) properties
MJAT_TS8_P1
Grade 12
Question:
Let $f(x)=x^3+6x^2-15x+3$. $f'(c)=0$ for some $c\in(a,b]$ with $f(a)=f(b)$. For maximum possible $b$: $g(x)=bx^3+ax^2+cx+d$ is decreasing in the largest interval $[e,f]$. The value of $3(e+f)$ is:
Step-by-Step Solution
Key Concept: $f'(x)=3x^2+12x-15=3(x+5)(x-1)$. Critical points at $x=-5$ and $x=1$. $c=-5$ or $1$. $f(a)=f(b)$ with $c\in(a,b]$: by Rolle's theorem. For max $b$: $f(a)=f(b)$ at a global level → $b=4$ (since $f(-5)=f(4)$? check). With $b=4$, $a=-5$, $c=1$.
$3(e+f)=\mathbf{2.50}$.
Correct Answer: 2.50