GATE 2018 EE Q.43: solution and answer

GATE 2018 · EEQ.43 · 2 marks · NAT

Let $f (x) = 3 x^{3} - 7 x^{2} + 5 x + 6$ . The maximum value of $f (x)$ over the interval $[0, 2]$ is ________ (up to 1 decimal place).

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Answer and solution

ANSWER

The maximum value of $f (x)$ on $[0, 2]$ is $12.0$ , attained at the endpoint $x = 2$ .

Official answer: 11.5 to 12.5Source: Official answer key, GATE 2018 EE (IIT Guwahati)

A continuous function on a closed interval reaches its absolute maximum either at a critical point inside the interval or at an endpoint (closed-interval method). $f (x)$ is a polynomial, so it is continuous on $[0, 2]$ , and we check both kinds of point.
Critical points: $f^{'} (x) = 9 x^{2} - 14 x + 5 = (9 x - 5) (x - 1) = 0 \Rightarrow x = \frac{5}{9}, x = 1$ . Both lie in $[0, 2]$ .
Second-derivative test: $f^{''} (x) = 18 x - 14$ . $f^{''} (\frac{5}{9}) = 10 - 14 = - 4 < 0$ , so $x = \frac{5}{9}$ is a local maximum. $f^{''} (1) = 4 > 0$ , so $x = 1$ is a local minimum.
Value at the local maximum: $f (\frac{5}{9}) = 3 \cdot \frac{125}{729} - 7 \cdot \frac{25}{81} + 5 \cdot \frac{5}{9} + 6 = \frac{125}{243} - \frac{525}{243} + \frac{675}{243} + \frac{1458}{243} = \frac{1733}{243} \approx 7.13$
Value at the local minimum: $f (1) = 3 - 7 + 5 + 6 = 7$ .
Endpoints: $f (0) = 6$ and $f (2) = 3 (8) - 7 (4) + 5 (2) + 6 = 24 - 28 + 10 + 6 = 12$ .
Compare all candidates: $max {6, 7.13, 7, 12} = 12$ , reached at the endpoint $x = 2$ . Maximum value $= 12.0$ .

KEY IDEA

For the absolute maximum on a closed interval, compare $f$ at every critical point inside the interval AND at both endpoints. A local maximum found with $f^{''} < 0$ is only a candidate. Here the cubic has a positive leading coefficient, so it is rising again after its local minimum at $x = 1$ and ends higher at $x = 2$ than at its local peak.

WATCH OUT FOR

Setting $f^{'} (x) = 0$ , finding the local maximum at $x = 5/9$ with $f^{''} < 0$ , and answering $f (5/9) \approx 7.1$ without checking the endpoints. The absolute maximum on $[0, 2]$ is at the endpoint $x = 2$ , where $f (2) = 12$ .

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GATE 2018 EE Q.43: solution and answer

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