MHT CET · Maths · Continuity and Differentiability

Let \(f: R \rightarrow R\) be a function defined as
\(f(x)=\left\{\begin{array}{ccc}
5 , & \text { if } x \leq 1 \
a+b x , & \text { if } 1 < x < 3 \
b+5 x,\end{array} \right. \) \(\text {if } 3 \leq x < 5 \
30 , \text {if } x \geq 5\) \(\mid \text {then } f \text { is }\)

A continuous if \(a=5\) and \(b=5\).
B continuous if \(a=-5\) and \(b=10\).
C not continuous for any values of \(a\) and \(b\).
D continuous is \(a=0\) and \(b=5\).

Verified Solution

Answer & Solution

Correct Answer

(C) not continuous for any values of \(a\) and \(b\).

Step-by-step Solution

Detailed explanation

for continuity at \(x=1, a+b=5\)

for continuity at \(x=3, a+3 b=b+15\)
\(\Rightarrow a+2 b=15\)

\(\because\) system of equation (i), (ii) and (iii) is inconsistent Hence, \(f(x)\) is not continuous for any values of \(a\) and \(b\) on \(R\)

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 \(f: R \rightarrow R\) be a function defined as\(f(x)=\left\{\begin{array}{ccc}5 , & \text { if } x \leq 1 \a+b x , & \text { if } 1 < x < 3 \b+5 x,\end{array} \right. \) \(\text {if } 3 \leq x < 5 \30 , \text {if } x \geq 5\) \(\mid \text {then } f \text { is }\)

Step-by-step Solution

Let \(f: R \rightarrow R\) be a function defined as
\(f(x)=\left\{\begin{array}{ccc}
5 , & \text { if } x \leq 1 \
a+b x , & \text { if } 1 < x < 3 \
b+5 x,\end{array} \right. \) \(\text {if } 3 \leq x < 5 \
30 , \text {if } x \geq 5\) \(\mid \text {then } f \text { is }\)