WBJEE · Maths · Differentiation

Let \(f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}\)
\(+x^{3}+x+19 .\) Then, \(f(x)=0\) has

A 13 real roots
B only one positive and only two negative real roots
C not more than one real root
D has two positive and one negative real root

Verified Solution

Answer & Solution

Correct Answer

(C) not more than one real root

Step-by-step Solution

Detailed explanation

We have, \(f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3}+x+19\) \[ \begin{array}{r} \Rightarrow f^{\prime}(x)=13 x^{12}+11 x^{10}+9 x^{4} \\ +7 x^{6}+5 x^{4}+3 x^{2}+1 \end{array} \] \(f^{\prime}(x)\) has no real root. \(\therefore f(x)=0\) has not more than one real root.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let \(f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}\) \(+x^{3}+x+19 .\) Then, \(f(x)=0\) has

Step-by-step Solution

Let \(f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}\)
\(+x^{3}+x+19 .\) Then, \(f(x)=0\) has