‹ Back to the paper
Study the Assertion and Reason and choose the correct option.
Let f(x) be a polynomial function of degree 7 such that d/(dx)(f(x)) = (x-2)³(x+1)²(7x-2) has a…
Assertion: Let $f(x)$ be a polynomial function of degree 7 such that $\frac{d}{dx}(f(x)) = (x-2)^3(x+1)^2(7x-2)$ has a local minimum at $x = -1$.
Reason: Let $f$ have first derivative at $c$ such that $f'(c) = 0$ and $f'(x) > 0, \forall x \in (c-\delta, c)$, $f'(x) < 0, \forall x \in (c, c+\delta)$, then $c$ is a point of local minimum.
Assertion (A): Let $f(x)$ be a polynomial function of degree 7 such that $\frac{d}{dx}(f(x)) = (x-2)^3(x+1)^2(7x-2)$ has a local minimum at $x = -1$.
Reason (R): Let $f$ have first derivative at $c$ such that $f'(c) = 0$ and $f'(x) > 0, \forall x \in (c-\delta, c)$, $f'(x) < 0, \forall x \in (c, c+\delta)$, then $c$ is a point of local minimum.
- (a)Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).
- (b)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
- (c)Assertion (A) is true, but Reason (R) is false.
- (d)Assertion (A) is false, but Reason (R) is true.
Answer
Answer
Correct option: (d)
Answer: (d) Assertion (A) is false, but Reason (R) is true.
A: $f'(x) = (x-2)^3 (x+1)^2 (7x-2)$. The factor $(x+1)^2$ never changes sign, so $f'$ does not change sign at $x = -1$ and there is no local extremum there: A is false.
R (as intended): $c$ is a local minimum when $f'$ changes from negative to positive at $c$ (the printed paper has the two signs the other way round, which would describe a local maximum). With that misprint corrected, R is true; this is the paper's key.