Functions
Critical and stationary points; false statement
Grade Class 12
Question:
Let $f(x)=x^7(x^3+2x^2-x-2)+(x^3-x)(x+3)+2(x^2-1)$. Which is NOT true?
$y=f(x)$ has 7 critical points
exactly 2 points in $(-1,1)$ where $f'(x)=0$
$y=f(x)$ does not have extrema at $x\in I$
$y=f(x)$ has exactly 2 stationary points in $(-2,1)$
Step-by-Step Solution
Key Concept: $f(x)=(x+2)(x+1)(x-1)(x^7+x+1)+2(x^2-1)$. $f'$ has degree 8 (up to 8 real roots). In $(-2,1)$: 3 stationary points (solution), so (D) is false.
(D) is NOT true.
Correct Answer: 4